Chapter 7: Statistical Analysis

Spreadsheet-Based Decision Support Systems

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Overview

7.1 Introduction 7.2 Understanding Data 7.3 Relationships in Data 7.4 Distributions 7.5 Summary

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Introduction

Performing basic statistical analysis of data using Excel functions

Statistical features of the Data Analysis Toolpack

Trend curves for analyzing data patterns

Basic linear regression techniques in Excel

Several different distribution functions in Excel

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Understanding Data

Statistical Functions

Descriptive Statistics

Histograms

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Statistical Functions

AVERAGE– Finds the mean of a set of data.– =AVERAGE(range or range_name)

MEDIAN– Finds the middle number in a list of sorted data.– =MEDIAN(range or range_name)

STDEV– Finds the standard deviation of a set of data.– This is equal to the square root of the variance, which measures the

difference between the mean of the data set and the individual values.– =STDEV(range or range_name)

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Figures 7.1 and 7.2

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Figures 7.3 and 7.4

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Analysis Toolpack

An Excel Add-In which includes several statistical analysis techniques

To ensure that it is an active Add-in, choose Tools > Add-ins from the menu. Select Analysis Toolpack from the list.

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Descriptive Statistics

Provides a list of statistical information about your data set including– Mean

– Median

– Standard deviation

– Variance

Go to Tools > Data Analysis > Descriptive Statistics

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Descriptive Statistics (cont)

The Input Range refers to the location of the data set.

You can check whether your data is Grouped By Columns or Rows.

If there are labels in the first row of each column of data, then check the Labels in First Row box.

The Output Range refers to where you want the results of the analysis to be displayed in the current worksheet.

The Summary Statistics box will calculate the most commonly used statistics from our data.

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Figure 7.7

Quarterly stock returns for three different companies are recorded. We want to know – Average stock return

– Variability of stock returns

– Which quarters had the highest and lowest stock returns

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Figures 7.8 and 7.9

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Figure 7.11

The standard deviation can be used to understand how common outliers are in the data.

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More Descriptive Statistics

Confidence Level for Mean– The mean is calculated using the specified confidence level (for example, 95% or

99%), the standard deviation, and the size of the sample data.– The confidence level and calculated mean are then added to the analysis report.– You can compare the actual mean to this calculated mean based on the specified

confidence level.

Kth Largest– Gives the largest ranked data value for a specified value of k.– For k = 1, the maximum data value would be returned.

Kth Smallest– Gives the smallest ranked data value for a specified value of k.– For k = 1, the minimum data value would be returned.

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Descriptive Statistics Functions

PERCENTILE – Returns a value for which a desired percentile k of the specified data_set falls below.– =PERCENTILE(data_set, k)

For example, for the MSFT data, the value for which 95% of the data falls below is

– =PERCENTILE(B4:B27,0.95) = 0.108

PERCENTRANK – Returns the percentile of the data _set which falls below a given value. – =PERCENTRANK(data_set, value)

For example, the percent of the MSFT data which falls below the value 0.108 is– =PERCENTRANK(B4:B27, 0.108) = 0.95, or 95%

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Histograms

Histograms calculate the number of occurrences, or frequency, which values in a data set fall into various intervals.

Choose the Histogram option from the Analysis Toolpack list.

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Histograms (cont’d)

The Input Range is the range of the data set.

The Bin Range is used to specify the location of the bin values.

– Bins are the intervals into which values can fall; they can be defined by the user or can be evenly distributed among the data by Excel.

The Output Range is the location of the output, or the frequency calculations for each bin.

The chart options include a simple Chart Output (the actual histogram), Cumulative Percentage for each bin value, and a Pareto organization of the chart.

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Figures 7.15 and 7.16

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Figures 7.17 and 7.18

To create your own bin values, make a list of upper bounds for each interval.

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Figure 7.19

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Histograms (cont’d)

Histograms can also be formatted.– Right-click on the histogram and change the Chart Options or other

parameters.

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Histograms (cont)

There are four basic shapes to a histogram: – Symmetric: has only one peak; that is, there is a central high part and almost

equal lower parts to the left and right of this peak.

– Positively skewed: has a peak on the left and many lower points (stretching) to the right.

– Negatively skewed: has a peak on the right and many lower points (stretching) to the left.

– Multiple peaks: imply that more than one source, or population, of data is being evaluated.

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Relationships in Data

Trend Curves

Regression

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Data Relationships

Relationships in data are usually identified by comparing two variables: the dependent variable and the independent variable.

– The dependent variable is the variable we are most interested in. By understanding its current behavior we can better predict its future behavior.

– The independent variable is the variable we use as the comparison in order to make this prediction.

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Trend Curves

Trend curves are used to graph and analyze these relationships between data.

Trend curves graph the data with – the independent variable on the x-axis

– the dependent variable on the y-axis

To add a trend curve to your chart, right-click on the data points in an XY Scatter chart and choose Add Trendline from the drop-down list of options.

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Trend Curves (cont’d)

There are five basic trend curves which Excel can model: – Linear

– Exponential

– Power

– Moving Average

– Logarithmic

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Trend Curves (cont’d)

Click on the Options tab to set options for the trend curve.

Set the name of the trendline.

Specify a period forward or backwards for which you want to predict the behavior of your dependent variable.

Check to Display Equation and Display R-Squared Value.

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Linear Trend Curves

Number of Units Produced each month and the corresponding Monthly Plant Cost are recorded.

The company wants to be able to estimate their plant costs based on the planned production amounts.

The dependent variable is therefore the Monthly Plant Cost and the independent variable is the Units Produced.

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Figures 7.26 and 7.29

Graph the data and then add a Linear trendline.

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Figure 7.30

Use the displayed equation to predict future values. First check the accuracy of the equation by calculating the error from the

known data.

Linear trends have the relationship: y = a*x - b

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Exponential Trend Curves

Sales data for ten years is recorded.

We want to be able to predict sales for the next few years.

The independent variable is Years and our dependent variable is Sales.

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Figures 7.34 and 7.35

Exponential trends have the relationship: y = a*e^(b*x) or

y = a*EXP(b*x)

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Power Trend Curves

We are given yearly Production values and yearly Unit Cost for production.

We want to determine the relationship between Unit Cost and Production in order to be able to predict future Unit Costs.

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Figures 7.39 and 7.40

Power trends have the relationship: y = a*x^b

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Regression Analysis

We can use some regression analysis parameters to ensure that the relationships we have chosen for our data are “good” fits.

These parameters include– R-Squared value

– Standard error

– Slope

– Intercept

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R-Squared Value

The R-Squared value measures the amount of influence the independent variable has on the dependent variable.

The closer the R-Squared value is to 1, the stronger the relationship is between the independent and dependent variables.

If the R-Squared value is closer to 0, then there may not be a relationship between these two variables.

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Figure 7.42

We fit a Linear trendline to the Monthly Plant Cost per Units Produced chart (see Figure 7.44).

The R-Squared value is 0.8137, which is fairly close to 1, implying a good fit.

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Figure 7.45

The RSQ Excel function can calculate the R-squared value from a set of data.– =RSQ(y_range, x_range)

Note that this function only works with Linear trend curves.

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Standard Error

The standard error measures the accuracy of any predictions made.

It can be calculated in Excel using the STEYX function– =STEYX(y_range, x_range)

This function can also only be used for Linear trend curves.

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Slope and Intercept

Two Excel functions can be used with a linear regression line of a collection of data.

SLOPE function– =SLOPE(y_range, x_range)

INTERCEPT function– =INTERCEPT(y_range, x_range)

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Distributions

Many distributions have Excel functions associated with them. – These functions are basically equivalent to using distribution tables.

– That is, given certain parameters of a set of data for a particular distribution, you would look at a distribution table to find the corresponding area from the distribution curve.

Some common distributions are– Normal

– Exponential

– Uniform

– Binomial

– Poisson

– Beta

– Weibull

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Normal Distribution

The parameters for this distribution are simply the value we are interested in finding the probability for, and the mean and standard deviation of the set of data.

The function we use with the Normal distribution is NORMDIST– =NORMDIST(x, mean, std_dev, cumulative)

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Normal Distribution (cont)

The cumulative parameter will be seen in many Excel distribution functions.

This parameter can take the values True or False to determine if you want the value returned from the cumulative distribution function or the probability density function, respectively. – The cumulative distribution function (cdf) will find the probability that a value

in the data set is less than or equal to x.

– The probability density function (pdf) will find the probability that a value is exactly equal to x.

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Figure 7.48

Annual drug sales at a local drugstore are distributed Normally with a mean of 40,000 and standard deviation of 10,000.

The probability that the actual sales for the year are 42,000 is 0.58, or 58%.

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Figure 7.49

What is the probability that annual sales will be between 35,000 and 49,000?

To find this value, we will subtract the cdf values for these two bounds.– =NORMDIST(49000, 40000, 10000, True) –

NORMDIST(35000, 40000, 10000, True) This will return a 0.51 probability, or 51% chance.

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Standard Normal Distribution

If the mean of your data is 0 and the standard deviation is 1, then placing these values in the NORMDIST function with the cumulative parameter as True will find the resulting value from the Standard Normal distribution.

The STANDARDIZE function will convert the x value from a data set of a mean not equal to 0 and a standard deviation not equal to 1 into a value which does assume a mean of 0 and a standard deviation of 1.– =STANDARDIZE(x, mean, std_dev)

The resulting standardized value is then used as the main parameter in the NORMSDIST function– =NORMSDIST(standardized_x)

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Figure 7.50

Consider the same example used previously to find the probability that a drugstore’s annual sales are 42,000.

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Uniform Distribution

The Uniform distribution does not actually have a corresponding Excel function; however, a simple formula can be used to model the Uniform distribution.– 1 / (b – a)

Given that a value x is Uniformly distributed between a and b, we can use this formula to determine the probability that x will have an integer value in this interval.

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Figure 7.51

Consider any values for a and b, then use the formula to calculate the Uniform value.

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Poisson Distribution

The Poisson distribution has only the mean as its parameter.

The function we use for this distribution is POISSON – =POISSON(x, mean, cumulative)

The Poisson distribution value is the probability that the number events which occur is either between 0 and x (cdf) or equal to x (pdf).

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Figure 7.52

For example, consider a bakery which serves an average of 20 customers per hour.

Find the probability that at most 35 customers will be served in the next two hours.

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Exponential Distribution

The Exponential distribution has only one parameter: lambda = 1 / mean of the data set.

The function we use for this distribution is EXPONDIST– =EXPONDIST(x, lambda, cumulative)

The Exponential distribution is commonly used for modeling interarrival times.

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Figure 7.53

Let us use the same example with the bakery data. Arrival rate is said to be 20 customers per hour. Interarrival mean, or the Exponential mean, is 1 / arrival rate. Therefore,

for this example, the interarrival mean is 1/20 hours per customer arrival.

To find the probability that a customer arrives in 10 minutes, we would set – x = 10/60 = 0.17 hours

– lambda = 1/(1/20) = 20 hours

– =EXPONDIST(0.17, 20, True)

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Binomial Distribution

The Binomial distribution has the following parameters: the number of trials and the probability of a success.

We are trying to determine the probability that the number of successes is less than or equal to (using cdf) or equal to (pdf) some x value.

The function we use for this distribution is BINOMDIST – =BINOMDIST(x, trials, prob_success, cumulative)

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Figure 7.54

Suppose a survey shows that 40 percent of people pay more attention to ads in the newspaper, and 60 percent pays more attention to ads on television.

What is the probability that out of 100 people surveyed, 50 of them respond more to ads on television?

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Beta Distribution

The Beta distribution has the following parameters: alpha, beta, A, and B. – Alpha and beta are determined from the data set

– A and B are optional bounds on the x value for which you want the Beta distribution value

The function we use for this distribution is BETADIST – =BETADIST(x, alpha, beta, A, B)

If A and B are omitted, then a standard cumulative distribution is assumed and they are given the values 0 and 1, respectively.

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Figure 7.55

Determine the probability that a team can complete a project in 10 days.

Estimate the total time needed to be 1 to 2 weeks; these estimates will be the bound values, or the A and B parameters.

Use a mean and standard deviation of 12 and 3 days to compute the alpha and beta parameters.

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Weibull Distribution

The Weibull distribution has the parameters alpha and beta.

The function we use for this distribution is WEIBULL – =WEIBULL(x, alpha, beta, cumulative)

The Weibull distribution is most commonly used to determine reliability functions.

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Figure 7.56

On average, a lightbulb will last 1200 hours, with a standard deviation of 100 hours. We can use these values to calculate alpha and beta.

We can now use the WEIBULL distribution to determine the probability that a lightbulb will be reliable for 55 days = 1320 hours.

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Summary

The Analysis Toolpack is an Excel Add-In that includes statistical analysis techniques such as Descriptive Statistics, Histograms, Exponential Smoothing, Correlation, Covariance, Moving Average, and others.

The Descriptive Statistics option provides a list of statistical information about a data set, including the mean, median, standard deviation, and variance.

Histograms calculate the number of occurrences, or frequency, which values in a data set fall into various intervals.

Relationships in data are usually identified by comparing the dependent variable and the independent variable.

There are five basic trend curves that Excel can model: Linear, Exponential, Power, Moving Average, and Logarithmic.

Some of the more common distributions that can be recognized when performing a statistical analysis of data are the Normal, Exponential, Uniform, Binomial, Poisson, Beta, and Weibull distributions.

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