# Overview

All tools have the same output options (see ). The results can be printed into a new sheet, into a new workbook, or into a given output range on a sheet of the current workbook. To select the output method select one of the radio buttons inside the Output frame. If you have chosen Output Range you must also enter a single range in the entry field.

Select the Autofit Columns option to automatically adjust the widths of the columns in the output range.

You will normally want to select the Clear Output Range option, since otherwise some of the cells with existing content will remain in the output range.

The Retain Output Range Formatting and Retain Output Range Comments options are useful if you have already preformatted the output range.

All analysis tools also provide a choice whether they will enter formulæ or just values in the cells. By default &gnum; will usually enter formulæ. These formulæ will automatically reevaluate when the data change. For some tools, the formulæ also permit modification of certain parameters.

If the chosen output range is too small, some of the results will be lost.

The old data in the output range is deleted and cannot be recovered.

## Common output options of the data analysis tools

An image of the output options dialog used by the statistical analysis tools.

To enter a range into an entry field, you can either type the range specification into the text field, or click in the text field and then select the range on the sheet (see ).

## Specifying Ranges

An image of the input range text box used by the statistical analysis tools.

Some entry fields accept lists of ranges. To enter these lists, select one range, type a comma, and then select the next range. At any time, you may switch to another sheet of the workbook.

# Descriptive Statistics

## Correlation Tool

### Correlation Tool Dialog

An image of the correlation analysis dialog.

The correlation tool calculates the pairwise Pearson correlation coefficients of the given variables. Use this tool to calculate any number of correlation coefficients at the same time. The variables for which the correlations are calculated are specified by the Input Range: entry. The input range can consist of either a single range or a comma separated list of ranges. The given range or ranges can be grouped by columns, by rows, or by areas.

If the first row or column of the given ranges, or the first field of each area contains labels, the Labels option should be selected.

### Some Example Data

An image of an example data set for a correlation analysis.

### Using the Correlation Tool

For example, you want to calculate the correlation between three variables, one each in columns A, B, and C. Both variables have 10 values in rows 2 to 11 with labels in row 1 (see ).

1. Enter A1:B11 in the Input Range: entry by typing this directly into the entry or clicking in the entry field and then selecting that range on the sheet. In the latter case the entry will also contain the sheet name.

2. Select the Columns radio button next to Grouped By:, since each variable is in its own column.

3. Select the Labels option since the first row contains labels. (see ).

4. Specify the output options as described above.

5. Press the OK button.

The calculated correlations are given in a table with each column and row labeled with the names of the variables. If the names are not given in the input range, &gnum; generates them. In our example, the correlation between the variables in column A and B, can be found in the second column and third row of the results table (see ).

### Completed Correlation Dialog

An image of the completed correlation analysis dialog.

### Correlation Tool Output

An image of the output of the correlation analysis.

## Covariance Tool

### Covariance Tool Dialog

An image of the covariance analysis dialog.

The covariance tool calculates the pairwise covariance coefficients of the given variables. Use this tool to calculate any number of covariance coefficients at the same time. The variables for which the covariances are calculated are specified by the Input Range: entry. The input range can consist of either a single range or a comma separated list of ranges. The given range or ranges can be grouped by columns, by rows, or by areas.

If the first row or column of the given ranges, or the first field of each area contains labels, the Labels option should be selected.

### Some Example Data

An image example data for a covariance analysis.

### Using The Covariance Tool

For example, you want to calculate the covariance between three variables, one each in columns A, B, and C. Both variables have 10 values in rows 2 to 11 with labels in row 1 (see ).

1. Enter A1:B11 in the Input Range: entry by typing this directly into the entry or clicking in the entry field and then selecting that range on the sheet. In the latter case the entry will also contain the sheet name.

2. Select the Columns radio button next to Grouped By:, since each variable is in its own column.

3. Select the Labels option since the first row contains labels.

4. Specify the output options as described above.

5. Press the OK button.

The calculated covariances are given in a table with each column and row labeled with the names of the variables. If the names are not given in the input range, &gnum; generates them. In our example, the covariance between the variables in column A and B, can be found in the second column and third row of the results table (see ).

### Covariance Tool Output

An image of the output of a covariance analysis.

## Descriptive Statistics Tool

### Descriptive Statistics Tool Dialog

An image of the descriptive statistics dialog.

The descriptive statistics tool calculates various statistics for the given variables and a confidence interval for the population mean. The variables are specified via the Input Range: entry. The given range or list of ranges can be grouped into variables by columns, rows, or areas.

This tool can produce four different kinds of statistical data.

- If the Summary Statistics option is selected, this tool calculates the mean, standard error, median, mode, standard deviation, sample variance, kurtosis, skewness, range, minimum, maximum, sum, and count for each variable.

- If the Confidence Interval for the Mean option is selected, the tool calculates confidence intervals for the population mean of each variable. Specify the confidence level in the entry box. The default confidence level is 95%.

  The interval given will usually be wider than the interval obtained using the CONFIDENCE function. The CONFIDENCE function assumes that the population standard deviation is known. This tool estimates the population standard deviation using the sample standard deviation.

- If the Kth Largest: option is selected, the tool finds the `k`th largest value of each of the variables. Specify `k` in the entry box next to the option. The default is 1.

- If the Kth Smallest: option is selected, the tool finds the `k`th smallest value of each of the variables. Specify `k` in the entry box next to the option. The default is 1.

If the first entry for each variable contains the label, select the Labels option.

### Some Example Data

An image of some example data for descriptive statistics.

### Using the Descriptive Statistics Tool

shows some example data, the selected options, and the corresponding output.

### The Options Page For Descriptive Statistics

An image of some example data for descriptive statistics.

### Descriptive Statistics Tool Output

An image of the output of a descriptive statistics analysis.

## Frequency Tables

&gnum; provides two types of frequencies tables:

- The frequency table tools is primarily useful for non-numeric data (data of nominal and ordinal level of measurement). It allows to determine frequencies for given values.

- The histogram tool is useful for numeric data that is supposed to be classified into a certain number of intervals. These intervals can be either specified or calculated.

### Frequency Tables Tool

#### Introduction

The frequency tool can be used to create frequency tables for non-numerical data. It presents this table numerically as well as graphically.

If your data are numeric and you want to accumulate whole intervals of values into frequency counts then this tool is not appropriate. In that case you may want to use the histogram table tool described in section .

##### Frequency Tool Dialog

An image of the dialog to generate various frequency tables open to the "Input" tab.

As shown in , the frequency table dialog has four tabs. We will introduce them in sequence.

#### The Input Tab

The Input tab shown in contains the field specifying the data to be used for the histogram.

The Input Range entry contains a single range or a list of ranges, that can be grouped into variables by rows, columns, or areas.

If the first row or column of the given input ranges, or the first field of each area contains labels, the Labels option should be selected. If the input is grouped by areas and the top left cell contains a label, the other cells in the first row are being ignored.

#### The Categories Tab

The Categories tab permits the specification of a range that contains the possible values that are supposed to be counted in the input range.

The automatic categories option is disabled since it is not yet implemented.

##### Frequency Tool Dialog Categories Tab

An image of the dialog to generate various frequency tables open to the "Categories" tab.

#### The Graphs & Options Tab

The Graphs & Options tab allows various options to be set. In the top half of the tab you can choose whether you would like a graph to be created. If you choose to have a graph created you can specify whether you would like to see a bar chart or a column chart.

In the bottom part of the tab you can select the percentages option. This option replaces the frequency counts with percentages.

If the categories range contains repeated values, then the percentages may add up to more than 100%. If the the categories range does not contain all values that occur in the input range, the percentages may sum to less than 100%.

The Use exact comparisons checkbox determines how category values and input range values are compared. If it is checked then the function `EXACT` is used for the comparison. If it isn't checked then simple equality is used. In this latter case, empty cells and cells containing the numerical value 0 are considered equal. As a consequence you usually want that checkbox to be selected.

##### Frequency Tool Dialog Graphs & Options Tab

An image of the dialog to generate various frequency tables open to the "Graphs & Options" tab.

#### Frequency Tool Results

##### Frequency Tool Results

Sample results of the frequencies tool.

### Histogram Tool

#### Introduction

The histogram tool can be used to create histograms or frequency tables for numerical data. Using this tool you can define intervals, or bins. The tool determines how many data points belong to each bin and presents this number numerically as well as graphically.

If your data are non-numeric this tool is not appropriate. In that case you may want to use the frequency table tool described in section .

##### Histogram Tool Dialog

An image of the dialog to generate various histograms open to the "Input" tab.

As shown in , the histogram dialog has five tabs. We will introduce them in sequence.

#### The Input Tab

The Input tab shown in contains the field specifying the data to be used for the histogram.

The Input Range entry contains a single range or a list of ranges, that can be grouped into variables by rows, columns, or areas.

If the first row or column of the given input ranges, or the first field of each area contains labels, the Labels option should be selected. If the input is grouped by areas and the top left cell contains a label, the other cells in the first row are being ignored.

#### The Cutoffs Tab

##### Histogram Tool Dialog Cutoffs Tab

An image of the dialog to generate various histograms open to the "Cutoffs" tab.

The cutoffs for the histogram can either be predetermined by data contained in your workbook or calculated by the histogram tool. These cutoffs determine bins as defined by the selection on the Bins tab.

Select the Predetermined Cutoffs option to specify data on your worksheet in the Cutoff Range: entry. The values in this range will be used as cutoffs `c1`, `c2`, and so on to `cn`.

Select the Calculated Cutoffs option to have the cutoffs determined by the tool. Enter the desired number of cutoffs in the Number of Cutoffs entry. It is strongly recommended (but optional) that you specify the minimum and maximum cutoffs in the Minimum cutoff and Maximum cutoff entries. If the minimum or maximum cutoff is not specified, the tool will use the minimum and/or maximum of the current data.

#### The Bins Tab

##### Histogram Tool Dialog Bins Tab

An image of the dialog to generate various histograms open to the "Bins" tab.

The bins tab is used to determine how the cutoffs `c1`, `c2`, and so on to `cn` are translated into bins. Specifically, it has to be determined whether first and/or last bins reaching from −∞ to `c1` and from `cn` to ∞ are added and whether data points that much cutoffs exactly are included in the bin to the right or the left.

For example the option [∙,∙),[∙,∙),⋯, [∙,∙),[∙,∞) indicates that the first bin starts at the first cutoff while the last bin ends at ∞. Moreover, each cutoff value belongs to the bin on its right.

#### The Graphs & Options Tab

##### Histogram Tool Dialog Graphs & Options Tab

An image of the dialog to generate various histograms open to the "Graphs & Options" tab.

The options in the graphs and options tab specify any graph to be created and modify the appearance of the histogram:

- The No chart option causes the chart to be omitted.

- The Bar chart option causes a bar chart to be added to the histogram. For each bin, the bar chart shows a horizontal bar indicating the frequency.

  The Column chart option causes a column chart to be added to the histogram. For each bin, the column chart shows a vertical bar indicating the frequency.

  The Histogram chart option causes a histogram chart to be added to the histogram. For each bin, the histogram chart shows a vertical bar indicating the density (that is the frequency divided by the width of the bin).

- The Percentages option causes the frequencies to be expressed as percentages.

- The Cumulative answers option causes a cumulative frequency table (either with counts or with pecentages) to be created.

- The Count numbers only option determines whether only numbers are counted. If also non-numbers are counted they are first converted into numbers, usually into 0.

#### The Output Tab

The Output tab contains the standard output options and fields described in .

#### A Histogram Example

##### Some Example Data

An image of some example data for use with the histogram tool.

##### Using the Histogram Tool

For example, you want to calculate a histogram for the number of successes in several sequences of trials. The numbers of successes are recorded in column A and the cutoffs of interest in column C (see ).

1. Enter A1:A31 in the Input Range: entry of the Input tab by typing this directly into the entry or clicking in the entry field and then selecting that range on the sheet. In the latter case the entry may also contain the sheet name.

2. Since you only have one variable select the Areas or Columns radio button next to Grouped By:.

3. Select the Labels option since the first cell of the Input Range contains a label.

4. Enter C2:C5 in the Cutoff Range: entry of the Cutoffs tab. The Predetermined Cutoffs option will now also be selected (see ).

5. In the Bins tab select the second option since we want to add two bins reaching to ∓∞ and we want to count each cutoff value in the bin to its right (see ).

6. Select the Percentage option of the Graphs &Options tab to have the frequencies expressed as percentages.

7. Select the Column Chart option of the Graphs &Options tab to have a column chart added to the histogram (see ).

8. In the Output tab, specify the output options as described in .

9. Press the OK button.

The results are shown in . Note that the graph will by default appear on top of the histogram table. It usually needs to be moved in to proper position. That has already been done here.

##### Histogram Tool: Specifying Cutoffs

An image of selecting the cutoffs for the example data used with the histogram tool.

##### Histogram Tool: Specifying Bins

An image of selecting a certain bins type.

##### Histogram Tool: Specifying Options

An image of specifying the required options.

##### Histogram Tool Output

An image of the output from the histogram analysis tool.

## Rank and Percentile Tool

### Rank and Percentile Tool Dialog

An image of the rank and percentile analysis tool.

Use this tool to rank given data and to calculate the percentiles of each data point.

Specify the datasets to use in the Input Range: entry. The given range can be grouped into datasets by columns, by rows, or by areas.

For each dataset, the tool creates three columns in the output table:

1. The first column gives the indices of the ordered data from largest to smallest data value.

2. The second column gives data values corresponding to the indices in the first column.

3. The third column indicates the percentile of the data value in the second column.

If you have labels in the first cell of each data set, select the Labels option.

### Some Example Data for the Rank and Percentile Tool

An image of example data for use with the rank and percentile analysis tool.

### Using the Rank and Percentile Tool

shows some example data and the corresponding output.

### Rank and Percentile Tool Output

An image of the output from a rank and percentile analysis.

In the case of ties, the rank calculated by this tool differs from the value of the RANK function for the same data. This tool calculates the rank as it is normally used in Statistics: If two values are tied, the assigned rank is the average rank for those entries. For example in the two values 10 are the second and third largest values. Since they are equal each receives the rank of 2.5, the average of 2 and 3. The rank function on the other hand assigns the rank as it is normally used to determine placements. The two values 10 would therefore each receive a rank of 2.

# Sampling Tool

## Sampling Tool Dialog

An image of the sampling tool.

Use the sampling tool to take a sample of a data set. This tool can take both a random sample of a given size or a periodic sample:

random sample

  A random sample is a subset of the population such that every subset of that size has the same chance of being picked.

periodic sample

  In a periodic sample every `k`th element in the population is selected.

To use this tool, first specify the data set or data sets by setting the Input Range: entry. The range or ranges given can be grouped into datasets by rows, by columns, or by areas.

If the first entry in each data set contains a variable, select the Labels option.

Select the sampling method which can be either periodic or random.

random sample

  Specify the size of the random sample in the Size of Sample: entry.

periodic sample

  Specify the period in the Period: entry.

Specify the number of samples you would like to obtain in the Number of Samples: entry.

Since the period uniquely determines a periodic sample, if you specify that you would like 2 samples you will be given the identical sample twice.

If the dataset for a periodic sample is a two dimensional range, &gnum; will enumerate the data points by row first.

## Some Example Data for the Sampling Tool

An image of example data for use with the sampling tool.

## Using the Sampling Tool

shows some example data and the corresponding output.

## Sampling Tool Output

An image of the output from the sampling tool.

# Dependent Observations

## Forecast Tools

### Exponential Smoothing Tool

#### Exponential Smoothing Tool Dialog

An image of the exponential smoothing dialog.

The Exponential Smoothing tool performs the exponential smoothing for the given set or sets of values. It provides the choice of 5 different exponential smoothing methods:

- Simple exponential smoothing according to (Hunter, 1968).

- Simple exponential smoothing according to (Roberts, 1959).

- Holt's trend corrected exponential smoothing (occasionally also referred to as double exponential smoothing)

- Additive Holt-Winters exponential smoothing

- Multiplicative Holt-Winters exponential smoothing (occasionally also referred to as triple exponential smoothing)

Since the kind of options available depend on the type of exponential smoothing desired, you can choose the type on the Input page.

#### Common Options of the Exponential Smoothing Tool

Specify the cells containing the datasets in the Input Range entry. The entered range or ranges are grouped into datasets either by rows or by columns.

If you have labels in the first cell of each data set, select the Labels option.

If you select the Include chart option, &gnum; will also create a chart showing both the data and corresponding smoothed values.

#### Exponential Smoothing According to Hunter

Each value in the smoothed set is predicted based on the forecast for the prior period. The formula is given in . α is the value given as Damping factor. `yt` is the `t`th value in the original data set and `lt` the corresponding smoothed value.

##### Exponential Smoothing Formula According To Hunter

The formula used in exponential smoothing according to Hunter.

For example, a value for α between 0.2 and 0.3 represents 20 to 30 percent error adjustment in the prior forecast.

If you choose to have the tool enter formulæ rather than values into the output region, then you can modify the damping factor α even after you executed the tool.

To have the standard errors output as well, check the Standard error check box. The formula used is given in . The denominator can be adjusted by selecting the appropriate radio button. Since there are `t−1` terms in the sum of the denominator, selecting n−1 means that the denominator will be `t−2`.

##### The Standard Error Formula For Exponential Smoothing According To Hunter

The formula used to calculate the standard error of exponential smoothing according to Hunter

If you check the Include chart check box, a line graph showing the observations `yt` and the predicted values `lt` will also be created.

##### Using the Exponential Smoothing Tool

shows some example data, the selected options and the corresponding output.

##### Some Example Data for the Exponential Smoothing Tool

An image of example data for exponential smoothing.

##### The Options for the Exponential Smoothing Tool

An image of the options tab of the exponential smoothing tool.

##### Exponential Smoothing Tool Output (Hunter)

An image of the output of an exponential smoothing analysis according to Hunter.

#### Exponential Smoothing According to Roberts

The simple exponential smoothing method according to Roberts is used for forecasting a time series without a trend or seasonal pattern, but for which the level is nevertheless slowly changing over time. The predicted values are calculated according to the formula given in . α is the value given as Damping factor. `yt` is the `t`th value in the original data set and `lt` the predicted value. `l0` is the predicted value at time 0 and must be estimated. This tool uses the average value of the first 5 observations as estimate.

If you choose to have the tool enter formulæ rather than values into the output region, then you can modify the damping factor α and the estimated value at time 0 after executing the tool.

##### Exponential Smoothing Formula According To Roberts

The formula used in exponential smoothing according to Roberts.

To have the standard errors output as well, check the Standard error check box. The formula used is given in . The denominator can be adjusted by selecting the appropriate radio button.

##### The Standard Error Formula For Exponential Smoothing According To Roberts

The formula used to calculate the standard error of exponential smoothing according to Roberts

If you check the Include chart check box, a line graph showing the observations `yt` and the predicted values `lt` will also be created.

##### Using the Exponential Smoothing Tool

shows example output for the exponential smoothing tool using the formula according to Roberts. Cell A4 contains the estimated level at time 0. If you requested to have formulæ rather than values entered into the sheet, then changing the estimate in A4 and/or the value for α in A2 will result in an immediate change to the predicted values.

##### Exponential Smoothing Tool Output (Roberts)

An image of the output of an exponential smoothing analysis according to Roberts.

#### Holt's Trend Corrected Exponential Smoothing

Holt's trend corrected exponential smoothing is appropriate when both the level and the growth rate of a time series are changing. (If the time series has a fixed growth rate and therefore exhibits a linear trend, a linear regression model is more appropriate.)

`yt` is the true value at time `t`, `lt` is the estimated level at time `t` and `bt` is the estimated growth rate at time `t`. We use the two smoothing equations given in to update our estimates. α is the value given as Damping factor and γ is the value given as Growth damping factor.

This tool obtains initial (time 0) estimates for the level and growth rate by performing a linear regression using the first 5 data values.

##### Formulae Of Holt's Trend Corrected Exponential Smoothing

The formulae used in Holt's trend corrected exponential smoothing.

If you choose to have the tool enter formulæ rather than values into the output region, then you can modify the damping factors α and γ as well as the estimated level and growth rate at time 0 after executing the tool.

To have the standard errors output as well, check the Standard error check box. The formula used is given in . The denominator can be adjusted by selecting the appropriate radio button.

##### The Standard Error Formula For Holt's Trend Corrected Exponential Smoothing

The formula used to calculate the standard error for Holt's trend corrected exponential smoothing.

If you check the Include chart check box, a line graph showing the observations `yt` and the estimated level values `lt` will also be created.

##### Using the Exponential Smoothing Tool

shows example output for Holt's trend corrected exponential smoothing. Cell A4 contains the estimated level at time 0 and B4 the estimated growth rate at time 0. If you requested to have formulæ rather than values entered into the sheet, then changing the estimates in A4, B4, the values for α in A2 and/or for γ in B2 will result in an immediate change to the predicted values.

##### Exponential Smoothing Tool Output (Holt's)

An image of the output of Holt's trend corrected exponential smoothing.

#### Additive Holt-Winters Method

The additive Holt-Winters method of exponential smoothing is appropriate when a time series with a linear trend has an additive seasonal pattern for which the level, the growth rate and the seasonal pattern may be changing. An additive seasonal pattern is a pattern in which the seasonal variation can be explained by the addition of a seasonal constant (although we allow for this constant to change slowly.)

`yt` is the true value at time `t`, `lt` is the estimated level at time `t`, `bt` is the estimated growth rate at time `t` and `st` is the estimated seasonal adjustment for time `t`. We use the three smoothing equations given in to update our estimates. α is the value given as Damping factor, γ is the value given as Growth damping factor and δ is the value given as Seasonal damping factor. `L` is the value given as Seasonal period. If your data consist of monthly values, then `L` should be 12, if it consist of quarterly values then `L` should be 4.

This tool obtains initial (time 0) estimates for the level and growth rate by performing a linear regression using all data values. It obtains estimates for the seasonal adjustments by averaging the appropriate seasonal differences from values predicted by linear regression alone.

##### Exponential Smoothing Formulae Of The Additive Holt-Winters Method

The formulae used in the additive Holt-Winters Method.

If you choose to have the tool enter formulæ rather than values into the output region, then you can modify the damping factors α, γ and δ as well as all estimates after executing the tool.

To have the standard errors output as well, check the Standard error check box. The formula used is given in . The denominator can be adjusted by selecting the appropriate radio button.

##### The Standard Error Formula Of The Additive Holt-Winters Method

The formula used to calculate the standard error in the additive Holt-Winters Method

If you check the Include chart check box, a line graph showing the observations `yt` and the estimated level values `lt` will also be created.

##### Using the Exponential Smoothing Tool

shows the options' tab of the exponential smoothing tool for the additive Holt-Winters method. The data is expected to have a seasonal period of 4 (this would for example happen if we have a data value for each quarter of a year). shows the corresponding example output for the additive Holt-Winters method. Cell C7 contains the estimated level at time 0, D7 the estimated growth rate at time 0, and E4 to E7 the initial seasonal adjustments for each of the 4 seasons preceding our data time period. If you requested to have formulæ rather than values entered into the sheet, then changing any of these estimates, the values for α in A2, for γ in B2 and/or for δ in C2 will result in an immediate change to the estimated values.

##### Exponential Smoothing Tool Options (Additive Holt-Winters))

An image of the options' tab for the additive Holt-Winters method.

##### Exponential Smoothing Tool Output (Additive Holt-Winters)

An image of the output of the additive Holt-Winters method.

#### Multiplicative Holt-Winters Method

The multiplicative Holt-Winters method of exponential smoothing is appropriate when a time series with a linear trend has a multiplicative seasonal pattern for which the level, the growth rate and the seasonal pattern may be changing. A multiplicative seasonal pattern is a pattern in which the seasonal variation can be explained by the multiplication of a seasonal constant (although we allow for this constant to change slowly.)

`yt` is the true value at time `t`, `lt` is the estimated level at time `t`, `bt` is the estimated growth rate at time `t` and `st` is the estimated seasonal adjustment for time `t`. We use the three smoothing equations given in to update our estimates. α is the value given as Damping factor, γ is the value given as Growth damping factor and δ is the value given as Seasonal damping factor. `L` is the value given as Seasonal period. If your data consist of monthly values, then `L` should be 12, if it consist of quarterly values then `L` should be 4.

This tool obtains initial (time 0) estimates for the level and growth rate by performing a linear regression using the data values of the first 4 seasonal periods. It obtains estimates for the seasonal adjustments by averaging the appropriate seasonal differences from values predicted by linear regression alone during the first 4 seasonal periods.

##### Exponential Smoothing Formulae Of The Multiplicative Holt-Winters Method

The formulae used in the multiplicative Holt-Winters Method

If you choose to have the tool enter formulæ rather than values into the output region, then you can modify the damping factors α, γ and δ as well as all estimates after executing the tool.

To have the standard errors output as well, check the Standard error check box. The formula used is given in . The denominator can be adjusted by selecting the appropriate radio button.

##### The Standard Error Formula Of The Multiplicative Holt-Winters Method

The formula used to calculate the standard error in the multiplicative Holt-Winters Method

If you check the Include chart check box, a line graph showing the observations `yt` and the estimated level values `lt` will also be created.

##### Using the Exponential Smoothing Tool

shows the example output for the multiplicative Holt-Winters method, assuming 4 seasons. Cell C7 contains the estimated level at time 0, D7 the estimated growth rate at time 0, and E4 to E7 the initial seasonal adjustments for each of the 4 seasons preceding our data time period. If you requested to have formulæ rather than values entered into the sheet, then changing any of these estimates, the values for α in A2, for γ in B2 and/or for δ in C2 will result in an immediate change to the estimated values.

##### Exponential Smoothing Tool Output (Multiplicative Holt-Winters)

An image of the output of the multiplicative Holt-Winters method.

### Moving Average Tool

#### Moving Average Tool Dialog

An image of the dialog for the moving average analysis tool.

Use the moving average tool to calculate moving averages of one or more data sets. A moving average provides useful trend information of the data that is lost in a simple average. In addition, moving averages can be used to eliminate random variance. For example, use this tool to create a smoother curve of a stock prize.

Specify the cells containing the datasets in the Input Range entry. The entered range or ranges are grouped into datasets either by rows or by columns.

If you have labels in the first cell of each data set, select the Labels option.

Choose the type of moving average you would like to calculate. The tool can determine 4 types of moving averages:

1. Simple moving average

2. Cumulative moving average

3. Weighted moving average

4. Spencer's 15 point moving average

#### Moving Average Tool Dialog: The Options Tab

An image of the Options tab of the moving average analysis tool.

Specify the Interval for the moving average. The interval `i` is the number of consecutive values to be included in each moving average. This options is only available for the simple and weighted moving averages.

Check the Standard errors checkbox if you would also like the standard error to be calculated. Since there is no general agreement on the denominator for the standard error you can choose the appropriate radio button.

In the case of the simple moving average, you can also choose between a prior moving average and a central moving average, or you may even specify any other desired offset.

1. Prior moving average: Each average takes into account the current observation and the most recent prior observations for a total of `i` observations.

2. Central moving average with `i` being odd: Each average takes into account the current observation and the same number of most recent prior observations and closest future observations for a total of `i` observations.

3. Central moving average with `i` being even: This is calculated according to the formula given in . `at` is the moving average at time `t` and `yt` is the observation at time `t`.

4. Other offset: If the offset is 0, this is just the prior moving average. Otherwise the offset indicates the number of closest future observations to include in the average. Correspondingly, the number of most recent past observations is decreased.

#### Formula For The Central Moving Average With Even Interval

The formula for the central moving average if the interval length is even.

The results are given in one column for each dataset (with a second column added if you have chosen standard errors to be calculated). Each row represents the moving average of the corresponding row or column in the input range. Depending on the type of average and the offset, the moving average cannot be calculated for the first rows in the input range.

#### Simple Moving Average

A simple moving average is the unweighted average of a collection of observations. Exactly which observations are included depends on whether a prior or central moving average is calculated.

#### Cumulative Moving Average

A cumulative moving average is a prior moving average in which the current and all prior observations are included.

#### Weighted Moving Average

A weighted moving average with an interval `i` is a prior moving average calculated according to formula . `at` is the moving average at time `t` and `yt` is the observation at time `t`.

##### Formula For The Weighted Moving Average With Interval `i`

The formula for the weighted moving average if the interval length is `i`.

#### Spencer's 15 Point Moving Average

Spencer's 15 point moving average is a central moving average calculated according to formula . `at` is the moving average at time `t` and `yt` is the observation at time `t`.

##### Formula For Spencer's 15 Point Moving Average

The formula for the Spencer's 15 point moving average.

#### A Moving Average Example

##### Some Example Data for the Moving Average Tool

An image of some example data for use with the moving average analysis tool.

##### Using the Moving Average Tool

shows some example data, shows the option settings, and the corresponding output.

##### Moving Averages Tool Options

An image of the option settings of the moving averages example.

##### Moving Averages Tool Output

An image of the output from the moving average analysis tool.

## Fourier Analysis Tool

### Fourier Analysis Tool Dialog

An image of the fourier analysis dialog.

The Fourier Analysis tool normally performs a Fast Fourier Transform to obtain the discrete fourier transform Fs of the given sequence ft of real numbers according to the formula given in .

Select the Inverse option to calculate the inverse discrete fourier transform ft of the given sequence Fs of real numbers

If the number of terms in the given sequence is not a power of 2 (i.e. 2, 4, 8, 16, 32, 64, 128, etc.), this tool will append zeros to reach such a power of 2!

Specify the cells containing the datasets in the Input Range entry. The entered range or ranges are grouped into sequences either by rows or by columns.

If you have labels in the first cell of each data set, select the Labels option.

### Fourier Analysis Formulae

The formulae used in a fourier analysis.

Before using the numbers obtained by this tool, ensure that these are in fact the correct formulae for your discipline. In the physical sciences this fourier transform tends to be called the inverse fourier transform and vice versa. Moreover, frequently the scaling factor varies.

For example Mathematica uses the terms fourier transform and inverse fourier transform with the reversed meaning than &gnum; and it uses a scaling factor of `1/SQRT(N)` rather than `1/N`.

## Kaplan Meier Estimates Tool

### The Input Tab

The Input tab shown in contains the fields specifying the data to be used for the Kaplan Meier Estimates. The time column contains the times or dates at which the subjects died or were censored. If any of the subjects were censored, the Permit censorship checkbox is checked and the Censor column contained the censorship marks. Censorship marks are typically 0s or 1s. The range of censor marks or labels can be set using the the remaining two spinboxes.

#### Kaplan-Meier Tool Dialog

An image of the .Kaplan-Meier tool dialog.

### The Groups Tab

If the subjects belong to several groups and the groups are supposed to be analyzed separately, the groups tab can be used.

#### Kaplan-Meier Tool Dialog Groups Tab

An image of the Kaplan-Meier tool dialog groups tab.

The groups tab can be enabled via the Define multiple groups checkbox. The groups column entry contains the address of the column specifying the group membership. Groups can then be defined or deleted via the Add and Remove buttons.

### The Options Tab

The options tab of the Kaplan-Meier tools dialog is used to set various options of the Kaplan-Meier tool.

#### Kaplan-Meier Tool Dialog Options Tab

An image of the Kaplan-Meier tool dialog options tab.

### The Output Tab

The Output tab contains the standard output options and fields described in .

### A Kaplan-Meier Example

#### Kaplan-Meier Tool Example Input

An image of the input to the Kaplan-Meier estimate example and of the input tab of the Kaplan-Meier analysis tool.

#### Using the Kaplan-Meier Tool

Suppose you want to calculate Kaplan-Meier Estimates for the as given in . Each row contains the data for one subject. Column A contains the survival time, i.e. the time until death or censure. Column B contains the group number, we are considering two groups of subjects. Column C indicates whether the subject died (0) or was censured (1).

We complete the fields of the Input tab as shown in . The time column is A2:A21 and the censure column is C2:C21.

Since we have two groups of subjects, on the Groups tab we check the Define multiple groups check box and set up two groups with identifiers 1 and 2 in column B2:B21:

#### Kaplan-Meier Tool Example Group Tab

An image of the group tab of the Kaplan-Meier analysis tool.

On the Options tab all checkboxes are pre-checked and we leave them that way to obtain a maximum amount of information.

On the output tab we choose where we would like the output to be placed. For the purposes of this example we retain the New Sheet target. After clicking OK we get the output shown in . Note that the graph initially always appears on top of the numerical result and was moved for the screen shot.

B1:F17 shows the results of the first group, G1 to K17 the results of the second group. The graph shows the Kaplan-Meier survival curves for both groups.

M4:N7 shows the result of the Mantel-Haenszel Log-Rank Test. In this case the p-value is larger than 0.3 and we would fail to reject the Null hypothesis. There is no evidence that the survival times differ.

#### Kaplan-Meier Tool Example Output

An image of the output of the Kaplan-Meier analysis tool.

## Principal Component Analysis

### Principal Component Analysis Tool Dialog

An image of the principal component analysis tool dialog.

Principal Component Analysis Tool performs a principal component analysis (PCA). PCA is a useful statistical technique with application in fields such as face recognition and image compression. It is a common technique for finding patterns in data of high dimension.

Specify the cells containing the datasets in the Input Range entry. The entered range or ranges are grouped into the factors either by rows or by columns.

If you have labels in the first cell of each factor, select the Labels option.

### Principal Component Analysis Example Data

An image of example data for use with the principal component analysis tool.

### Using the Principal Component Analysis Tool Tool

Suppose you want to perform a principal component analysis on the data given in having the two dimensions (factors) x and y.

1. Enter Sheet1!$A$1:$B$11 (or just A1:B11) in the Input Range: entry by typing this directly into the entry or clicking in the entry field and then selecting the range on the sheet.

2. Select the Labels option since the first row contains labels. (see ).

3. Specify the output options as described above.

4. Press the OK button.

The output of this principal component analysis is shown in . The output shows the covariance matrix, the eigenvalues and corresponding eigenvectors. The principal component is the constructed factor with the highest percent of trace, ξ1.

### Principal Component Analysis Tool Output

An image of the output from a principal component analysis.

## Regression Tool

### Regression Tool Dialog

An image of the regression tool dialog.

The regression tool performs a multiple regression analysis.

Enter a range or list of ranges containing the independent variables into the X Variables: entry.

Enter a single range containing the dependent variable into the Y Variable: entry.

If the ranges for the independent and dependent variables also contains labels in the first field of each row, column or area, select the Labels option.

Specify the confidence level in the Confidence Level: entry. The default is 95%.

To force the regression line or plane to pass through the origin, select the Force Intercept To Be Zero option.

Specify the output options as described above. If the output is directed into a specific output range, that range should contain at least seven columns and 17 rows more than there are independent variables.

### Regression Example Data

An image of example data for use with the regression tool.

### Using the Regression Tool

Suppose you want to perform a regression analysis on the data given in using `v` and `y` as independent variables and `u` as dependent variable.

1. Enter B1:C11 in the X Variables: entry by typing this directly into the entry or clicking in the entry field and then selecting the range on the sheet.

2. Enter A1:A11 in the Y Variable: entry.

3. Select the Labels option since the first row contains labels. (see ).

4. Specify the output options as described above.

5. Press the OK button.

The output of this regression analysis is shown in .

### Completed Regression Dialog

An image of the regression tool dialog with the required fields completed.

### Regression Tool Output

An image of the output from a regression analysis.

# One Sample Tests

## Normality Tests

The normality test tool provides for four tests of normality.

1. Anderson Darling Test

2. Cramér-von Mises Test

3. Lilliefors (Kolmogorov-Smirnov) Test

4. Shapiro-Francia Test

### Normality Test Dialog

An image of the normality test dialog.

The data range is specified via the Input Range: entry (see ). The given range or list of ranges can be grouped into separate data sets by columns, rows, or areas. The tool performs a separate test for each data set.

### Test Tab of the Normality Test Dialog

An image of the test tab of the normality test dialog.

On the test tab one specifies which of the four tests to perform, the significance level for the test and whether to include a normal probability plot of the data (see ).

### Normality Test Example Data

An image of example data for a normality test.

### Using the Normality Test Tool

Suppose you want to perform a Lilliefors (Kolmogorov-Smirnov) Test for Normality on the data given in .

1. Enter A1:A50 (or Sheet1!$A$1:$A$50) in the Input Range: entry by typing this directly into the entry or clicking in the entry field and then selecting the range on the sheet.

2. Select the Labels option since the first row contains a label. (see ).

3. On the test tab of the dialog (see ) select the Lilliefors (Kolmogorov-Smirnov) Test.

4. Specify an appropriate significance level Alpha, say 0.05.

5. Select the Create Normal Probability Plot option to include a normal probability plot in the output.

6. Specify the output options as described above.

7. Press the OK button.

The output of this normality test is shown in . Note that the graph appears initially on top of the output data and needs to be moved to make the data visible.

### Completed Input Tab of the Normality Test Dialog

An image of the completed input tab of the normality test dialog.

### Completed Test Tab of the Normality Test Dialog

An image of the completed test tab of the normality test dialog.

### Normality Test Output

An image of the output from a normality test.

## One Median

The One Median test tool provides two non-parametric tests that test the null hypothesis that the sample comes from a population with a given median:

1. Sign Test

2. Wilcoxon Signed Rank Test

Selecting the appropriate submenu item opens the dialog with the respective test preselected.

### Sign Test

This section describes the one sample sign test to test the null hypothesis that the sample comes from a population with the given median. The tool to perform a sign test to test the null hypothesis that two paired samples come from populations with the same median is in section .

#### One-Median Test Dialog

An image of the one-median test dialog used by the Sign Test and the Wilcoxon Signed Rank Test.

The Sign Test tool performs a one-sample sign test whether the sample comes from a population with a given median.

The sample data range is specified via the Input Range: entry (see ). The given range or list of ranges can be grouped into separate data sets by columns, rows, or areas. The tool performs a separate test for each data set.

On the Testtab of the dialog (see ) the predicted median as well as the significance level are specified.

#### The Test Tab of the One-Median Test Dialog

An image of the test tab of the one-median test dialog.

#### Using the Sign Test Tool

Suppose you want to perform a Sign Test on the data given in to determine whether the sample comes from a population of mean 3.

1. Enter A1:A19 (or Sheet1!$A$1:$A$19) in the Input Range: entry by typing this directly into the entry or clicking in the entry field and then selecting the range on the sheet.

2. Select the Labels option since the first row contains a label. (see ).

3. On the Test tab of the dialog (see ) select the Sign Test.

4. Specify an appropriate significance level Alpha, say 0.05.

5. Select thepecify the median of the null hypothesis (3) in the Predicted Median entry.

6. Specify the output options as described above.

7. Press the OK button.

The output of this sign test is shown in .

#### Output of a Sign Test

An image of the output of a Sign Test.

### Wilcoxon Signed Rank Test

This section describes the one sample Wilcoxon signed rank test to test the null hypothesis that the sample comes from a population with the given median. The tool to perform a Wilcoxon signed rank test to test the null hypothesis that two paired samples come from populations with the same median is in section .

The Wilcoxon Signed Rank TTest tool performs a one-sample sign test whether the sample comes from a population with a given median.

The sample data range is specified via the Input Range: entry (see ). The given range or list of ranges can be grouped into separate data sets by columns, rows, or areas. The tool performs a separate test for each data set.

On the Testtab of the dialog (see ) the predicted median as well as the significance level are specified.

The p-values given by this tool are determined using a normal approximation. This approximation is only valid if the sample size is at least 12.

#### Using the Wilcoxon Signed Rank Test Tool

Suppose you want to perform a Wilcoxon Signed Rank Test on the data given in to determine whether the sample comes from a population of mean 3.

1. Enter A1:A19 (or Sheet1!$A$1:$A$19) in the Input Range: entry by typing this directly into the entry or clicking in the entry field and then selecting the range on the sheet.

2. Select the Labels option since the first row contains a label. (see ).

3. On the Test tab of the dialog (see ) select the Wilcoxon Signed Rank Test.

4. Specify an appropriate significance level Alpha, say 0.05.

5. Select thepecify the median of the null hypothesis (3) in the Predicted Median entry.

6. Specify the output options as described above.

7. Press the OK button.

The output of this sign test is shown in .

#### Output of a Wilcoxon Signed Rank Test

An image of the output of a Wilcoxon Signed Rank Test.

# Two Sample Tests

## Comparing Means of Two Populations

&gnum; provides four similar tools to test whether the difference of two population means is equal to a hypothesized value. These four tools use the same dialog (see ).

### `t`- and `z`-Test Tool Dialog

An image of the t-test and z-test dialog.

Depending on the options settings, the appropriate test will be performed. The entries in the Input, Test, and Output frames are independent from the specific test.

Enter the first variable in the Variable 1 Range entry and the second variable in the Variable 2 Range entry.

Enter the hypothesized difference between the population means in the Hypothesized Mean Difference entry, which has a default of 0. Enter the significance level in the Alpha entry, which has a default of 5 %.

Specify the output options as described above. If the output is printed into a range, it should have at least three columns and ten rows.

There are up to three possible options that can be selected:

Paired versus Unpaired

  If the variables are dependent (or paired) select the Paired option.

Known versus Unknown

  For unpaired or independent variables, the population variances may be known or unknown. In the latter case they will be estimated using the sample variances. Select the Known option if you in fact know the population variances prior to collecting the sample.

Equal versus Unequal

  For paired variables with unknown population variances, we may either assume that the population variances are equal or not. If the population variances are assumed to be equal, &gnum; will estimate the common variance by pooling the sample variances. Select the Equal option to assume that the population variances are equal.

### `t`-Test: Paired Two Sample for Means Tool

#### `t`-Test (Paired) Tool Dialog Options

An image of the options for the t-test.

For paired variables, when you click on OK, &gnum; will test whether the mean of the difference between the paired variables is equal to the given hypothesized mean difference.

#### Using the `t`-Test (Paired) Tool

See for an example of a completed dialog and for the corresponding output.

#### `t`-Test (Paired) Example Data

An image of the example for a t-test.

#### Output from the `t`-Test (Paired) Tool

An image of the output results from a t-test.

### `t`-Test: Two-Sample Assuming Equal Variances Tool

#### `t`-Test (Equal Variances) Tool Dialog Options

An image of the options for a t-test analysis of two samples with equal variances.

For unpaired variables with unknown but assumed equal population variances, when you click on OK, &gnum; will test whether the mean of the difference between the paired variables is equal to the given hypothesized mean difference.

#### Using the `t`-Test (Unknown but Equal Variances) Tool

See for an example of a completed dialog and for the corresponding output.

#### `t`-Test (Unknown but Equal Variances) Example Data

An image of example data for use with a t-test with unknown but equal variances.

#### Output from the `t`-Test (Unknown but Equal Variances) Tool

An image of the output from a t-test with unknown but equal variances.

### `t`-Test: Two-Sample Assuming Unequal Variances Tool

#### `t`-Test (Unknown and Unequal Variances) Tool Dialog Options

An image of the options in a t-test of two samples with unknown and possibly unequal variances.

For unpaired variables with unknown and assumed unequal population variances, when you click on OK, &gnum; will test whether the mean of the difference between the paired variables is equal to the given hypothesized mean difference.

#### Using the `t`-Test (Unknown and Unequal Variances) Tool

See for an example of a completed dialog and for the corresponding output.

#### `t`-Test (Unknown and Unequal Variances) Example Data

An image of example data for use in a t-test of two samples with unknown and possibly unequal variances.

#### Output from the `t`-Test (Unknown and Unequal Variances) Tool

An image of the output of a t-test of two samples with unknown and possibly unequal variances.

### `z`-Test: Two Samples for Means Tool

#### `z`-Test Tool Dialog Options

An image of the options in a z-test of two samples.

For unpaired variables with known population variances, enter those variances in the Variable 1 Pop. Variance and Variable 2 Pop. Variance entries. When you click on OK, &gnum; will test whether the mean of the difference between the paired variables is equal to the given hypothesized mean difference.

#### Using the `z`-Test Tool

See for an example of a completed dialog and for the corresponding output.

#### `z`-Test Example Data

An image of example data for use in a z-test of two samples.

#### Output from the `z`-Test Tool

Output from the `z`-Test Tools
An image of the output from a z-test of two samples.

## Comparing Medians of Two Populations

&gnum; provides three non-parametric tests to test the null hypothesis that the two samples come from populations with the same median. Two tests, performed through the same tool, apply in the case of paired samples:

- Sign Test

- Wilcoxon Signed Rank Test

One test applies in the case of unpaired samples:

- Wilcoxon-Mann-Whitney Test

### Sign Test

This section describes the two sample (paired) sign test to test the null hypothesis that the two samples come from populations with the same median. The tool to perform a sign test to test the null hypothesis that the single sample comes from a population with a given median is in section .

This section needs to be written.

### Wilcoxon Signed Rank Test

This section describes the two sample (paired) Wilcoxon signed rank test to test the null hypothesis that the two samples come from populations with the same median. The tool to perform a Wilcoxon signed rank test to test the null hypothesis that the single sample comes from a population with a given median is in section .

This section needs to be written.

### Wilcoxon-Mann-Whitney Test

This section needs to be written.

## F-Test: Two-Sample for Variances Tool

### F-Test Tool Dialog

An image of the dialog for an F-test analysis of the equality of two variances.

Use the F-Test tool to test whether two population variances are different against the null hypothesis that they are not.

Specify the variables in the Variable 1 Range: and Variable 2 Range: entries. The Alpha: entry contains the significance level which is by default 5%.

If the first field of each range contains labels, select the Labels option. The names of the variables will be included in the output table.

The results are given in a table. This table contains the mean, variance, count of observations and the degree of freedom for both variables. The output table also includes the F-value, the one-tailed probability for the F-value, and the F Critical value for one-tailed test and the corresponding values for a two tailed test. The one-tailed probability for the F-value (P(F≤f) one-tail row) is the probability of making a Type I error in the one-tailed test. Similarly, the two-tailed probability for the F-value (P two-tail row) is the probability of making a Type I error in the two-tailed test. Since in the two-tailed F-Test both critical values are positive, the F Critical two-tail row contains two numbers.

If the output is directed into a specific output range, that range should contain at least three columns and eight rows.

### Some Example Data

An image of some example data for an F-test of the equality of two variances.

### Using the F-Test Tool

shows some example data and the corresponding output.

### F-Test Tool Output

An image of the output of an F-test analysis of the equality of two variances.

# Multiple Sample Tests

## Analysis of Variance

### ANOVA: Single Factor Tool

Use this tool to perform a single factor analysis of the variances of given variables. The variables are specified by the Input Range: entry. The given range can be grouped into the variables either by columns, by rows or by areas. The Alpha: entry specifies the significance level which is by default 5%.

If the first row or first column of the given range, or the first field of each area contains labels, select the Labels option. The names of the variables will be included in the output table.

The results of this analysis of variance are presented in a standard ANOVA table. The F critical value is the largest value of F that is statistically significant using the given significance level (Alpha).

This tool also calculates the count, sum, average, and the variance of each variable.

#### 1-factor ANOVA Dialog and Example Data

An image of a multilevel single factor ANOVA analysis.

#### Using the single factor ANOVA

See for an example of a completed dialog and for the corresponding output.

#### Output From a 1-factor ANOVA

An image of the output from a multilevel single factor ANOVA analysis.

### ANOVA: Two-Factor Tool

&gnum; can perform two factor fixed effects ANOVAs with and without replication. The same dialog is used and the appropriate tool is selected depending on whether the number of rows per sample is 1 or larger than 1.

#### ANOVA: Two-Factor Without Replication Tool

If the number of rows per sample is given as 1, &gnum; performs a two factor fixed effects ANOVA without replication. Each column of the input range is interpreted as a level of the first factor while each row is interpreted as a level of the second factor.

The first row and column of the range may contain labels for these levels. In this case the Labels option should be selected.

The Alpha: entry specifies the significance level which is by default 5%.

##### Using the 2-factor ANOVA Without Replication Tool

See for an example of a completed dialog and for the corresponding output.

##### 2-factor ANOVA Without Replication Dialog

An image of a two factor ANOVA without replication analysis.

##### Output From a 2-factor ANOVA Without Replication

An image of the output from a two factor ANOVA without replication analysis.

#### ANOVA: Two-Factor With Replication Tool

If the number of rows per sample is larger than 1, &gnum; performs a two factor fixed effects ANOVA with replication. Each column of the input range is interpreted as a level of the first factor while groups of rows (the number of rows in each group given by the number of rows per sample value) are interpreted as levels of the second factor.

The first row and column of the range may contain labels for these levels. In this case the Labels option should be selected.

The Alpha: entry specifies the significance level which is by default 5%.

See for an example of a completed dialog and for the corresponding output.

##### 2-factor ANOVA With Replication Dialog

An image of a two factor ANOVA with replication analysis.

##### Output From a 2-factor ANOVA With Replication

An image of the output from a two factor ANOVA with replication analysis.

&gnum; will estimate missing values for each level combination as the mean of the existing values in that combination. The degrees of freedom are adjusted appropriately.

## Tests for a Contingency Table

### Test of Homogeneity

### Test of Independence