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# Regression Analysis and Best Fit Lines (XE0124)

Article ID: 103839

Article Last Modified on 6/8/2005

APPLIES TO

- Microsoft Excel 97 Standard Edition
- Microsoft Excel 98 for Macintosh

This article was previously published under Q103839

## SUMMARY

The Application Note "Regression Analysis and Best Fit Lines" (XE0124), discusses how to use Microsoft Excel functions to perform simple, multiple, and polynomial regression analysis. It contains examples of how to use LINEST(), LOGEST(), TREND(), and GROWTH() to describe a best fit line or curve and to make predictions about your data. It also outlines some of the new statistical functions and tools available with Microsoft Excel versions 4.0 and later.
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### Text of Xe0124

The following is the full text of the Application Note. (Note that some graphics and formatting may be missing from this version of the document. For best results, download the Microsoft Word version of this document.)

                                     OVERVIEW
                                     ========

    This Application Note discusses how to use Microsoft Excel functions to
    perform simple, multiple, and polynomial regression analysis. It contains
    examples of how to use LINEST, LOGEST, TREND, and GROWTH to describe a
    best-fit line or curve and to make predictions about your data. It also
    outlines some of the built-in statistical functions and tools available
    with Microsoft Excel.

       General Information
          To Use the Analysis ToolPak
          Choosing the Best Function
             When Your Data Is Linear
             When Your Data Is Exponential
             When Your Data Is Curvilinear

       Simple Regression Analysis
          Describing a Best-fit Line
             Finding the Slope and the Y Intercept
          Calculating a Best-fit Line
             Using TREND
             Using LINEST
          Plotting the Best-fit Line
             Plotting the Trendline Automatically
          Predicting Future Values
             Using FORECAST
             Using TREND
             Using LINEST

       Multiple Regression Analysis
          Predicting Y-Values
             Using TREND
             Using LINEST

       Polynomial Regression Analysis
          Calculating a Polynomial Curve
          Charting a Polynomial Curve

       Using Regression Statistics
             Using LINEST/LOGEST for Regression Statistics
             Using R2 to Test Regression Model Accuracy

    GENERAL INFORMATION
    ===================

    Regression is a statistical method used to predict values based on
    relationships in existing data. By analyzing how a single dependent
    variable (y) is affected by the values of one or more independent variables
    (x), you can predict what y will be given x. You can use this information
    to fit a line or a curve to your existing data and to forecast future
    values. The LINEST, TREND, LOGEST, and GROWTH functions are the primary
    functions you will use to perform regression analysis in Microsoft Excel.

    While this Application Note focuses primarily on the functions that can be
    used in Microsoft Excel versions 3.0 and later, Microsoft Excel versions
    4.0 and later offer several new functions and tools that you can use to
    perform regression analysis and to create best-fit lines. When one of these
    new functions can be used to perform a task described in this Application
    Note, the function will be noted in the appropriate section. The following
    table lists some of these new functions.

                                            Use this
       To do this                           function
       ----------------------------------------------

       Return the correlation coefficient   CORREL
       for two arrays of cells

       Return a single predicted y-value    FORECAST
       based on a linear regression of
       known x and y ranges

       Return the y intercept of the linear INTERCEPT
       regression line

       Calculate R2, the coefficient of     RSQ
       Determination

       Return the slope of the linear       SLOPE
       regression line

       Return the standard error of the     STEYX
       Regression

       Table 1--Regression Analysis Functions in
       Microsoft Excel Versions 4.0 and later.

    In addition, the Analysis ToolPak add-in provides a special set of
    analysis tools, including tools to accomplish the following tasks.

                                                Use this
       To do this                               analysis tool
       -------------------------------------------------------

       Predict a value based on the forecast    Exponential
       for the prior period, adjusted for the   Smoothing
       error in that prior forecast

       Project values in the forecast period    Moving Average
       based on the average value of the
       variable over a specific number of
       preceding periods

       Perform linear regression analysis and   Regression
       return statistics and plots as specified

                   Table 2--Analysis ToolPak Add-in Features

    To Use the Analysis ToolPak
    ---------------------------

    In Microsoft Excel 5.0 and later:

    1. On the Tools menu, click Data Analysis.

    2. If the Data Analysis command is not available, click Add-Ins on the
       Tools menu. In the Add-Ins dialog box, click to select the Analysis
       ToolPak check box.

    NOTE: If the Analysis ToolPak add-in is not listed, run the Setup program,
    choose Add/Remove, and select the Add-ins option for Microsoft Excel.

    In Microsoft Excel 4.0:

    1. On the Options menu, click Analysis Tools.

    2. If the Analysis Tools command is not available, click Add-In on the
       Options menu. In the Add-Ins dialog box, click Add. Click Analysis.xla
       in the Library\Analysis folder.

    3. In the Data Analysis dialog box, choose the tool that you want to use,
       such as Exponential Smoothing. For help on how to use a particular
       analysis tool, click Help in the dialog box for the tool.

    CHOOSING THE BEST FUNCTION
    ==========================

    Whether you are performing simple regression (one x variable), multiple
    regression (two or more x variables), or polynomial regression (one x
    variable raised to different powers), you will get the most accurate
    results if the function that you choose to regress your data is based on
    the patterns in your existing data.

    When Your Data Is Linear
    ------------------------

    Your data is linear if the rate of change in your data is even to such an
    extent that when you plot it in a chart, the pattern in your data points
    resembles a line. If your data is linear, use the linear regression
    functions, LINEST and TREND. Both functions use the "least squares" method
    to calculate a straight line that best fits your data. LINEST returns
    information about the line, such as its slope and y intercept, and TREND
    returns predicted values along the line.

    In Microsoft Excel versions 4.0 and later, the Regression tool (in the
    Analysis ToolPak add-in) performs linear regression, returns regression
    statistics, calculates best-fit lines, and creates best-fit line charts.

    When Your Data Is Exponential
    -----------------------------

    Your data is exponential if the rate of change in your data, when plotted
    on a chart, resembles a curve that rises or falls at an increasingly higher
    rate. If your data is exponential, use the logarithmic regression
    functions, LOGEST and GROWTH. LOGEST calculates an exponential curve that
    best-fits your data and, like LINEST, returns information about the curve.
    Like TREND, GROWTH returns predicted values along the curve.

    When Your Data Is Curvilinear
    -----------------------------

    To most accurately predict values when the pattern in your data is neither
    linear nor exponential, use polynomial regression in conjunction with the
    TREND function to calculate a best-fit curve. For example, use this method
    if, when you plot your data in a chart, it resembles a curve for which the
    rate of change is not dramatic or if your data fluctuates in such a way
    that no linear or curved pattern can be identified.

    SIMPLE REGRESSION ANALYSIS
    ==========================

    Your regression analysis is "simple" if you have only one independent x
    variable for each dependent y variable. For example, assume you are
    analyzing the sales figures for the first six months of operation for
    Wingtip Toys, a company that specializes in the design and manufacture of
    toys.

       NOTE: The following examples primarily use the LINEST and TREND
       functions. Wherever these two functions are discussed, LOGEST and
       GROWTH can be substituted if your data is exponentially curved and
       if a curve fit would be more accurate than a straight line.

    In the following sample data, the values in the Month column are the
    independent x variables and the values in the Sales column are the
    dependent y variables. Based on this data, you can describe, calcu late,
    and plot a best-fit line, and you can then predict future sales figures.
    Because the data is linear, you will use the LINEST and TREND functions to
    perform the regression analysis.

    The Regression tool in Microsoft Excel version 4.0 and later performs each
    of these tasks automatically.

    For additional information on calculating regression, see the following
    references.

       Version of
       Microsoft Excel   Reference
       ------------------------------------------------------------

       97, 98            In Help, search for "Regression, Analysis"
       7.0               In Help, search for "Regression"
       5.0               In Help, search for "Regression "
       4.0               User's Guide 2, pages 41-45

    Because this tool performs linear regression, if your data resembles
    an exponential curve, use LOGEST and GROWTH.

    Following are the sales figures for Wingtip Toys and the corresponding
    months in both table and chart form.

             A       B
       1   Month   Sales
       2        1  $4,200
       3        2  $6,100
       4        3  $7,300
       5        4  $7,300
       6        5  $8,700
       7        6  $10,500

             Table 3--Sample Data (Sales Figures for Wingtip Toys)

    DESCRIBING A BEST-FIT LINE
    ==========================

    The equation of a straight line is y=mx+b, where m is the slope and b is
    the y intercept. LINEST returns the slope (m) and y intercept (b) values
    that describe the line derived from your existing data.

    Microsoft Excel versions 4.0 and later provide specific SLOPE and INTERCEPT
    functions for calculating the slope and the y intercept when your data is
    linear.

    For additional information, see the following references.

       Version of
       Microsoft Excel   Reference
       ------------------------------------------------------------

       97, 98            In Help, search for "slope" or "intercept"
       7.0               In Help, search for "slope" or "intercept"
       5.0               In Help, search for "slope" or "intercept"
       4.0               Function Reference Guide, pages 405-406

       NOTE: If your data is exponentially curved, use LOGEST to return
       the slope (m) and y intercept (b) values that describe the curve.
       The equation used by LOGEST is y=b*m^x.

    Finding the Slope and the Y Intercept
    -------------------------------------

    To calculate the values of the slope (m) and y intercept (b), use the
    procedure appropriate for your version of Microsoft Excel.

    Microsoft Excel 4.0 and later:

    1. Use the data in Table 3--Sample Data (Sales Figures for Wingtip Toys).

    2. To find the slope, select cell E2 and type the following formula:

          =SLOPE(B2:B7,A2:A7)

    The slope of the line for this data is 1122.857.

    3. To find the y intercept, select cell F2, and type the following
       formula:

          =INTERCEPT(B2:B7,A2:A7)

    The point at which the line crosses the y axis is 3420.

    Microsoft Excel 3.0:

    1. Using the data in "Table 3--Sample Data (Sales Figures for Wingtip
       Toys)," select cells E2:F2.

    2. Type the following formula:

          =LINEST(B2:B7,A2:A7)

       NOTE: Because the function returns data to more than one cell, you
       must enter the formula as an array by pressing CTRL+SHIFT+ENTER in
       Microsoft Excel for Windows or COMMAND+RETURN in Microsoft Excel
       for the Macintosh.

    The first argument in the LINEST function is the array containing the known
    y-values (which in this example are the Sales numbers). The second argument
    is the array containing the known x-values (in this case, the Month
    numbers).

       NOTE: LINEST also takes other optional arguments that are not
       necessary for this example.

    The result 1122.857, in E2, is the slope, and the result 3420, in cell
    F2, is where the line crosses the y-axis (y intercept).

            E        F
       1  Slope      Y intercept
       2  1122.857    3420

       Table 4--Example of Slope Intercept Values

    CALCULATING A BEST-FIT LINE
    ===========================

    If your data is linear, use TREND or LINEST to calculate your best-fit
    line. In Microsoft Excel versions 4.0 and later, you can also use the
    FORECAST function (forecast is mainly useful for finding a data point based
    on existing data, but can it also be used as a substitute for the TREND
    function). If your data fits an exponential curve, use LOGEST or GROWTH.

    Using TREND
    -----------

    The TREND function is the easiest and most efficient function for
    calculating the points along a best-fit line. To simultaneously
    calculate all the values on the best-fit line, do the following:

    1. Using the data in Table 3 of this Application Note, select cells C2:C7
       and type the following formula:

          =TREND(B2:B7,A2:A7)

       NOTE: Because the function returns data to more than one cell, you
       must enter the formula as an array by pressing CTRL+SHIFT+ENTER in
       Microsoft Excel for Windows or COMMAND+RETURN in Microsoft Excel
       for the Macintosh.

    The first argument in the TREND function is the array containing the known
    y-values and the second argument is the array containing the known x-
    values.

       NOTE: TREND also takes other optional arguments that aren't
       necessary for this example.

    Using LINEST
    ------------

    You can also use the slope (m) and the y intercept (b) values returned by
    LINEST to find data points on the best-fit line by substituting the y-
    values or the x-values into the equation for a line (y=mx+b). By plugging
    each month number into this formula, you can calculate all the data points
    for your best-fit line. With the slope(m) value (1122.857) in cell E2 and
    the intercept (b) value (3420) in cell F2, do the following to generate the
    points on your best-fit line:

    1. Select cell D2 and enter the following formula (because the formula
       only references single cells, it is not necessary to enter this formula
       as an array):

          =($E$2*A2)+$F$2

    2. Select cells D2:D7.

    3. Use the method appropriate for you version of Microsoft Excel.

        - In Microsoft Excel 5.0 and later, point to Fill on the Edit menu,
          and click Down.

        - In Microsoft Excel 4.0 and earlier, click Fill Down on the Edit
          menu.

    The values returned are the y-values for your best-fit line.

    The following table shows the results of the values returned when you
    use TREND and LINEST.

            A        B          C             D           E           F
       -----------------------------------------------------------------------
       1  Month    Sales   Predicted Y   Predicted Y    Slope    Y Intercept
                              TREND         LINEST
       -----------------------------------------------------------------------
       2       1    $4,200       $4,543        $4,543  1122.857          3420
       3       2    $6,100       $5,666        $5,666
       4       3    $7,300       $6,789        $6,789
       5       4    $7,300       $7,911        $7,911
       6       5    $8,700       $9,031        $9,031
       7       6   $10,500      $10,157       $10,157

      Table 5--Results of the Values Returned When You Use TREND and LINEST.

       NOTE: The returned values for TREND and LINEST are identical.

    PLOTTING THE BEST-FIT LINE
    ==========================

    Once you have calculated the values on your best-fit line, you can add that
    line to your existing chart by copying cells C1:C7 and pasting them into
    your existing chart.

    The resulting chart will have a straight line (best-fit line) running
    through your original data.

       NOTE: When you display the points of a best-fit line against the
       original data in a chart, in most cases you will get the best results
       by using an xy (scatter) chart. If you use a line chart, the x-values
       will be treated as labels rather than as values, and curved lines may
       result.

    Plotting the Trendline Automatically
    ------------------------------------

    In Microsoft Excel versions 5.0 and later, you can insert a trendline
    directly into your chart without having to first calculate the points or
    copy and past those points into the chart. Do the following to
    automatically insert a trendline in a chart:

    1. Double-click the chart to activate it.

    2. Select the series for which you want to plot a trendline.

    3. Use the procedure appropriate for your version of Microsoft Excel:

       If you are using Microsoft Excel 97 or 98, click Add Trendline on the
       Chart menu. In the Trendline dialog box, click the Type tab, and then
       select the type of trend or regression you want to plot. Click OK.

       If you are using Microsoft Excel 5.0 or 7.0, click Trendline on the
       Insert menu.

    PREDICTING FUTURE VALUES
    ========================

    In addition to returning values along the line fitted to your existing
    data, you can use TREND and LINEST to predict future values. In Microsoft
    Excel 4.0 and later, you can also use the FORECAST function to predict
    future values.

    Using the Wingtip Toys example, suppose you want to calculate sales figures
    for months 7, 8, and 9. The following examples show how to accomplish this
    using the FORECAST, TREND, and LINEST functions, respectively.

    To predict values for months 7, 8, and 9, first enter the month
    numbers for which you want predicted sales figures and then use the
    FORECAST function to calculate the values.

    Using FORECAST
    --------------

    1. In cells A8:A10, type 7, 8, and 9, respectively.

    2. Select cells B8:B10.

    3. Type the following formula:

          =FORECAST(A8:A10,B2:B7,A2:A7)

       NOTE: Because the function returns data to more than one cell, you
       must enter the formula as an array by pressing CTRL+SHIFT+ENTER in
       Microsoft Excel for Windows or COMMAND+RETURN in Microsoft Excel
       for the Macintosh.

    The first argument in the FORECAST function is the array containing
    the new x-values for which you want to derive predicted y-values.

    The resulting values in cells B8:B10 are the predicted sales for the
    next three months.

             A       B
       1   Month   Sales
       2     1     $4,200
       3     2     $6,100
       4     3     $7,300
       5     4     $7,300
       6     5     $8,700
       7     6     $10,500
       8     7     $11,280
       9     8     $12,403
       10    9     $13,526

       Table 6--Sample Data Using TREND to Predict Future Values Using TREND

    1. In cells A8:A10, type 7, 8, and 9, respectively.

    2. Select cells B8:B10.

    3. Type the following formula:

          =TREND(B2:B7,A2:A7,A8:A10)

       NOTE: Because the function returns data to more than one cell, you must
       enter the formula as an array by pressing CTRL+SHIFT+ENTER in Microsoft
       Excel for Windows or COMMAND+RETURN in Microsoft Excel for the
       Macintosh.

    The third argument in the TREND function is the array containing the
    new x-values for which you want to derive predicted y-values.

    The resulting values in cells B8:B10 are the predicted sales for the
    next three months.

             A       B
       1   Month   Sales
       2     1     $4,200
       3     2     $6,100
       4     3     $7,300
       5     4     $7,300
       6     5     $8,700
       7     6    $10,500
       8     7    $11,280
       9     8    $12,403
       10    9    $13,526

           Table 7--Sample Data Using TREND to Predict Future Values

       NOTE: In Microsoft Excel versions 4.0 and later, you can use the
       AutoFill feature to predict future values. Using the data in the
       previous table, if you wanted to predict sales for months 7, 8, and 9,
       you would select cells B2:B7, select the AutoFill handle in the lower-
       right corner of the selected area, and drag down three additional
       cells. (The AutoFill method is by far the easiest method to use for
       predicting values; however, if you use the formulas, it is easier to
       tell which values are derived and which values are static). The
       functions give you more power and flexibility than the AutoFill feature
       does.

       CAUTION: In addition to returning predicted values for months 7, 8, and
       9, the data in cells B2:B7 will be overwritten with the values that
       represent the best-fit line. If you do not want your original data to
       be overwritten, copy it to a separate area on your worksheet and then
       use AutoFill.

    Using LINEST
    ------------

    To obtain the new y-values, you can also substitute the slope and y
    intercept values that you derived with the LINEST function (these results
    are on page 5) and the new x-values (7, 8, and 9) into the formula, y=mx+b.
    See Using LINEST in the "Calculating a Best-fit Line" section for step-by-
    step instructions on how to do this.

                           MULTIPLE REGRESSION ANALYSIS
                           ============================

    When you have two or more independent x variables for each y variable, the
    regression analysis is considered multiple. For example, you could predict
    a child's weight given his or her age and height. Assume you've collected
    the following data

            A      B      C
       1   Age   Height  Weight
       2       3     32     35
       3       5     40     40
       4       6     39     43
       5      10     50     70

            Table 8--Sample Data: Age, Height, and Weight of Child

    where the values under Weight (C2:C5) represent the dependent y variables
    and the values under Age and Height (A2:B5) represent the independent x
    variables.

    PREDICTING Y-VALUES
    ===================

    You can use either the TREND or the LINEST function to analyze the
    relationship of the age and height to weight, and you can make
    predictions based on the results of this analysis.

    In Microsoft Excel 4.0 and later, the Regression tool can also be used
    to predict y-values in a multiple regression model.

       NOTE: Do not use the FORECAST function because it only works for
       simple regression.

    For additional information on predicting values, see the following
    references.

       Version
       Of Microsoft Excel   Reference
       ---------------------------------------------------------------
       97, 98               In Help, search for "Multiple, Regression"
       7.0                  In Help, search for "Multiple Regression"
       5.0                  In Help, search for Multiple Regression"
       4.0                  User's Guide 2, pages 41-45

    Using TREND
    -----------

    Using TREND to Predict a Child's Weight:

    This example uses the data in Table 8--Sample Data: Age, Height, and
    Weight of Child.

    To use TREND to predict the weight of a 9-year-old, 45-inch child, do
    the following:

    1. In cells A6 and B6, type 9 and 45, respectively.

    2. Select cell C6 and type the following formula:

          =TREND(C2:C5,A2:B5,A6:B6)

       Because the function returns data to more than one cell, you must enter
       the formula as an array by pressing CTRL+SHIFT+ENTER in Microsoft Excel
       for Windows or COMMAND+RETURN in Microsoft Excel for the Macintosh.

    The result of the formula, 63.42, is the predicted weight.

    Using LINEST
    ------------

    To predict a y-value with LINEST, you must first calculate the slopes for
    each x variable and find the y intercept. Because a slope is returned for
    each x variable, when you use the LINEST function, you must first select a
    range of cells that consist of a single row and a single column plus an
    additional column for each x variable in your data table. In this example,
    because you have two x variables, you will need to select a range of three
    cells, three columns wide by one row tall.

    Using LINEST to Calculate a Child's Weight:

    This example uses the data in Table 8--Sample Data: Age, Height, and
    Weight of Child.

    To calculate the slopes and the y intercepts, select cells A7:C7 and
    type the following formula:

       =LINEST(C2:C5,A2:B5)

    NOTE: Because the function returns data to more than one cell, you must
    enter the formula as an array by pressing CTRL+SHIFT+ENTER in Microsoft
    Excel for Windows or COMMAND+RETURN in Microsoft Excel for the Macintosh.

    The following values will be returned

            A     B      C
       7  -0.32  5.98     24

    where -0.32 is the slope for the second x variable (height), 5.98 is
    the slope for the first x variable (age), and 24 is the y intercept.

       NOTE: The slopes are in reverse order: the first slope value
       corresponds to the second x variable and the second slope value
       corresponds to the first x variable. The LINEST function always
       returns the slopes in reverse order when more than one x variable
       is involved.

    You can use the slope values and the y intercept value to make
    predictions based on your data. Using the formula,
    y=(m1*x1)+(m2*x2)+(mn*xn)+b, you can predict the weight of a 9-year
    old, 45-inch child:

        =(9*5.98)+(45*-0.32)+24

    The result of the formula, 63.42, is the predicted weight.

    Similar to LINEST, the Regression tool in Microsoft Excel versions 4.0 and
    later returns the slope values and the y intercept value. As described
    previously, you can plug these values into the formula, y=mx+b, to predict
    y.

       NOTE: Because the x variables are independent, there may not be a
       good graphical representation for a multiple regression model. Each
       x-value can be plotted with its corresponding y-value, but the
       individual lines may be completely unrelated, and, therefore, may
       be meaningless.

                          POLYNOMIAL REGRESSION ANALYSIS
                          ==============================

    When your data is neither exponentially curved nor consistently
    linear, use the polynomial method of regression. When you plot a best-
    fit curve calculated with polynomial regression, the curve will rise
    and fall with the data.

    CALCULATING A POLYNOMIAL CURVE
    ==============================

    To calculate a polynomial curve, the dependent y variable is regressed
    against the independent x variable raised to different powers. To
    illustrate this process, take the following example. Fitting a straight
    line to the following data would not accurately predict the sales for any
    given month.

             A       B
       1   Month   Sales
       2        1  $4,200
       3        2  $1,600
       4        3  $5,120
       5        4  $4,500
       6        5  $5,400
       7        6  $1,460

       Table 9--Six-Month Sales Figures Wingtip Toys

    In this case, you will get the best results by setting up the following
    polynomial regression model.

           A    B    C      D      E        F
       10  X   X^2  X^3    X^4   Sales    Trend
       11   1     1    1      1  $4,200   $4,089
       12   2     4    8     16  $1,600   $2,154
       13   3     9   27     81  $5,120   $4,011
       14   4    16   64    256  $4,500   $5,609
       15   5    25  125    625  $5,400   $4,846
       16   6    36  216   1296  $1,460   $1,571

       Table 10--Polynomial Regression Model for Wingtip Sales Figures

    The values in cells A11:A16 are the month numbers copied from cells A2:A7
    of Table 9--Six-Month Sales Figures Wingtip Toys. The values in cells
    B11:D16 are the original x variables raised to the second, third, and
    fourth powers, respectively. To obtain these values, do the following:

    1. Select cell B11 and enter the formula:

          =A11^2

    2. Select cell C11 and enter the formula:

          =A11^3

    3. Select cell D11 and enter the formula:

          =A11^4

    4. Select cells B11:D16.

    5. Use the method appropriate for your version of Microsoft Excel.

        - In Microsoft Excel version 5.0 or later, point to Fill on the Edit
          menu, and click Down.

        - In Microsoft Excel version 4.0 or earlier, click Fill Down on the
          Edit menu.

    The values in E11:E16 are the sales figures copied from B2:B7. To derive
    the trend values in column F, select cells F11:F16 and type the following
    formula:

          =TREND(E11:E16,A11:D16)

       NOTE: Because the function returns data to more than one cell, you
       must enter the formula as an array by pressing CTRL+SHIFT+ENTER in
       Microsoft Excel for Windows or COMMAND+RETURN in Microsoft Excel
       for the Macintosh.

    CHARTING A POLYNOMIAL CURVE
    ===========================

    To add the TREND results to this chart, select cells F10:F16, and then
    copy and paste them into your existing chart.

                            USING REGRESSION STATISTICS
                            ===========================

    The LINEST and LOGEST functions can return additional regression
    statistics that can be helpful in using and evaluating your regression
    model. If you are using Microsoft Excel 4.0 or later and have linear
    data, you can use the Regression Tool from the Analysis ToolPak add-
    in. This tool will automatically return all the regression statistics.
    If your data resembles an exponential curve, use LOGEST to return
    accurate regression statistics.

    USING LINEST/LOGEST FOR REGRESSION STATISTICS
    =============================================

    To return the additional statistics using LINEST or LOGEST, you must
    select a range that includes five rows and a single column plus an
    additional column for each x variable in your data. In addition, the
    stats argument, which is the fourth argument in both of these
    functions, must be set to TRUE.

    The following table lists the ages, weights, and heights of a number
    of children.

            A      B      C
       1   Age   Height  Weight
       2       3     32     35
       3       5     40     40
       4       6     39     43
       5      10     50     70

            Table 11--Sample Data: Age, Height, and Weight of Child

    To return the additional regression statistics using the data from "Table
    11--Sample Data: Age, Height, and Weight of Child" use the following steps:

    1. Select cells D1:F5.

    NOTE: This range consists of five rows and a single column plus two
    additional columns (one for each x variable).

    2. Type the following formula:

          =LINEST(C2:C5,A2:B5,,TRUE)

    Because the function returns data to more than one cell, you must enter
    the formula as an array by pressing CTRL+SHIFT+ENTER in Microsoft Excel
    for Windows or COMMAND+RETURN in Microsoft Excel for the Macintosh.

    The resulting data should resemble the data in the following table.

              D          E         F
       1  -0.32      5.98      24
       2  2.243569   5.647619  57.68449
       3  .950813    6.024948  #N/A
       4  9.665289   1         #N/A
       5  701.7      36.3      #N/A

       Table 12--Regression Statistics

    The first row of the statistics contains the slope for the height, the
    slope for the age, and the y intercept. The second row contains the
    standard error of the slopes and of the y intercept. The third row contains
    R2 and the standard error for the y estimate. The fourth row contains the F
    statistic and degrees of freedom. And, the fifth row contains the
    regression sum of squares and the residual sum of squares.

    USING R2 TO TEST REGRESSION MODEL ACCURACY
    ==========================================

    A particularly useful statistic returned is the coefficient of
    determination called R2. In Microsoft Excel versions 4.0 and later, you can
    also use the RSQ function to find R2. This R2 indicator ranges in value
    from 0 to 1 and reveals how closely the estimated y-values correlate to
    your actual y-values. The closer R2 is to 1, the more perfect the
    correlation-this correlation indicates that the regression equation is very
    useful in accurately predicting a y-value. On the other hand, the closer R2
    is to 0, the less helpful it will be in predicting a y-value.

    In the previous example, the value for R2 returned by LINEST is .95, an
    excellent correlation. This indicates that, based on the collected data,
    the LINEST model can be used to make extremely accurate predictions of a
    child's weight given a specific age and height.

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