# Chi-Square PopulationVariance Test in Excelfor Marketing

It’s hard to predict exactly how much each one of your new customers will buy, but you probably have a good idea of the range that you would expect your customer’s spending to fall within.

What if you suddenly noticed a blip in recent customer order amounts which indicated that your customer spending spread might be widening? Is there a way know for sure whether the spending spread really has widened, or is this just a temporary aberration that grabbed your attention but may not mean anything?

In statistical terms, the question you would be asking is: Has the standard deviation of my customers’ order size increased? Good news --> There is a convenient statistical test you can quickly run in Excel to find that out. The test is called the Chi-Square Variance Test and is used to determine if the variance of a population has changed. Variance equals the standard deviation squared so if a population’s standard deviation increases, so does its variance, to an even greater degree.

The Chi-Square Variance Test is a great and simple way to determine whether your customers are more or less focused in their purchases of you products. More importantly, the Chi-Square Variance Test tells you whether your customers are being affected by something is changing what they buy from your company.

If variance of your customers’ spending has become smaller, your customer order size has become more predictable and more focused. If the variance has increased, your customer order size has become less predictable and less focused.

This Test Will Not Tell You What Changed, Only That Something Has Changed.

If the range (standard deviation) of customer spending on individual orders changes, the product mix that your customers normally purchase is changing. The underlying reason for this probably has important implications for the marketing program. If standard deviation (and therefore the variance) of order size changes, you will want to investigate further and find out why.

Here is a video that will demonstrate step-by-step how to perform the Chi-Square Variance Test in Excel to determine if the majority of your customers really have become more or less focused in their spending on individual orders.

Here is a Step-By-Step Video Showing How to Find Out If Your Customers Have Become More or Less Focused In Their Spending By Using the Chi-Square Variance Test in Excel:
(Is Your Sound Turned On?)

What Is the Chi-Square Variance Test?

The Chi-Square Variance Test and is used to determine if the variance of a population has changed. Marketers use the Chi-Square Variance Test to determine if the expected range of customer spending is changing. If so, something is affecting the buying habits of the customers.

The Chi-Square Variance Test consists of just 2 calculations that require only 4 inputs total.

These 4 inputs are:

1) Historic Standard Deviation, σ, of the population – This would be the long-time standard deviation of customer spending per order. It shouldn’t be too hard to calculate a standard deviation of past customer order size. Population Standard Deviation is usually denoted as σ, sigma.

2) Standard Deviation, s, of a recent large (at least 30) sample drawn randomly from the population. Make sure that the sample is random and is representative of the population from which the Population Standard Deviation was taken. Sample Standard Deviation is usually denoted as s.

3) The Sample Size, n.

4) The Degree of Certainty desired in the test. For example, you might want to be at least 95% certain of the outcome determined by the test.

The Chi-Square Variance Test requires measurements of standard deviation, not variance. That has no effect because, as mentioned above, variance is derived from standard deviation. Variance equals standard deviation squared.

The 5 Steps of the Chi-Square Variance Test

There are 5 steps in the Chi-Square Variance Test. They are;

Step 1) Determine the Required Level of Certainty, and, therefore, α, Alpha.

Step 2) Measure Sample Standard Deviation (s) from a large recent random sample drawn from the same population from which the Population Standard Deviation (σ) was derived. Sample size, n, must be at least 30.

Step 3) Calculate the Chi-Square Statistic.

Chi-Square Statistic = [ (n-1)*(s*s) ] / [σ*σ]

Step 4) Calculate the Curve Area Outside of the Chi-Square Statistic.

There are 2 possibilities:

a) If Sample Standard Deviation, s, is greater than the population Standard Deviation (σ):

Calculate the Area in the Right Outer Tail to the Right of the Chi-Square Statistic by this formula:

Tail Area Right of Chi-Square Statistic =
CHIDIST( Chi-Square Statistic, n-1 )

In this blog article and attached video, we will color the tail area outside the chi-Square Statistic with RED.

We will also color the area under the curve that represents alpha with yellow, as follows:

Click on the Image To See a Larger Version

The Red Area Outside the Chi-Square Statistic
(Is Smaller Than the Yellow Alpha Area)
Click on the Image To See a Larger Version

b) If Sample Standard Deviation, s, is less than the population Standard Deviation, σ,

then: Calculate the Area in the Left Outer Tail to the Left of the Chi-Square Statistic Tail:

Tail Area Left of Chi-Square Statistic =
1 - CHIDIST( Chi Square Statistic, n-1 )

The 5% Alpha Area (Yellow) Resulting
From the 95% Required Certainty

Click on the Image To See a Larger Version

The Red Area Outside the Chi-Square Statistic
(Is Larger Than the Yellow Alpha Area)
Click on the Image To See a Larger Version

The area under the Chi-Square curve that lies outside of the Chi-Square Statistic is sometimes called the P Value. For example, if 3% of the curve area lies outside the Chi-Square Statistic, then the P Value is 0.03.

Step 5) Analyze Using the Chi-Square Statistic Rule: If the area under the curve outside the Chi-Square Statistic is less than alpha, the population variance has moved in the direction of Sample Standard Deviation.

For example, if alpha is 0.05 (you require 95% certainty and alpha is therefore 0.05) and only 3% of the area under the Chi-Square curve lies outside of the Chi-Square Statistic, then you can now state with 95% certainty that the variance had moved.

The variance would have moved in the direction of the Sample Standard Deviation. If Sample Standard Deviation was measured to be greater the Population Standard Deviation and the curve area outside of the Chi-Square Statistic (3% = 0.03) was less than alpha (0.05), you can state with 95% certainty that population variance has increased.

To sum it up with charts:

If the red area fits inside the yellow area, we can state with the required degree of certainty that the population variance has moved in the direction of the sample standard deviation. In the case directly below, the red area does fit inside the yellow (alpha) area so we can state that the population varianced moved to the right (increased), with 95% certainty:

The Red Area Outside the Chi-Square Statistic
Is Smaller Than the Yellow Alpha Area

Click on the Image To See a Larger Version

If the red area DOES NOT fit inside the yellow area, we CANNOT state with the required degree of certainty that the population variance has moved in the direction of the sample standard deviation. In the case directly below, the red area DOES NOT fit inside the yellow (alpha) area so we CANNOT state that the population varianced moved to the left (decreased), with 95% certainty:

The Red Area Outside the Chi-Square Statistic
Is Larger Than the Yellow Alpha Area

Click on the Image To See a Larger Version

Here is the Test We Ran

Here is the problem definition: Customers on a commercial web site have historically had a standard deviation of 1.6 in the number of items they buy on individual purchase orders. The company’s Internet marketing manager took a random sample of 50 recent orders and measured the standard deviation of that sample to be 1.9 items per order.

The Internet marketing manager wanted to know with at least 95% certainty whether the population standard deviation had increased (had moved in the direction of the sample standard deviation).

The Required 4 Pieces of Information

1) Population Standard Deviation, σ, of item per order = 1.6

2) Sample Standard Deviation, s, of items per order = 1.9

3) Sample size, n = 50

4) Required Level of Certainty = 95%

The 5 Steps of the Chi-Square Variance Test

Using the 5-Step Chi-Square Variance Process, the Internet Marketing Manager determines within 95% certainty whether the population variance has increased as follows.

Step 1)
Determine the Required Level of Certainty, and, therefore α, Alpha.

The Required Level of Certainty is 95%. Alpha, α, is 0.05.
Alpha = 1 – Required Level of Certainty = 1 – 95% = 0.05

Step 2)
Measure Sample Standard Deviation (s) from a recent, large, representative random sample drawn from the same population from which the Population Standard Deviation (σ) was derived.
Sample Standard Deviation, s, of items per order = 1.6.  Population Standard Deviation, σ, of item per order = 1.6

Step 3)
Calculate the Chi-Square Statistic
Chi-Square Statistic = [ (n-1)*(s*s) ] / [σ*σ]Chi-Square Statistic = [ (50 - 1) * (1.9 * 1.9) ] / [1.6 * 1.6] = 69.09766

Step 4)
Calculate the Curve Area Outside of the Chi-Square Statistic

If Sample Standard Deviation, s, is greater than the population Standard Deviation (σ):

then calculate the Area in the Right Outer Tail outside of the Chi-Square Statistic by this formula:

Tail Area Right of Chi-Square Statistic =

CHIDIST( Chi-Square Statistic, n-1 )

Since s is greater than σ,

Tail Area Right of Chi-Square Statistic =
CHIDIST( Chi-Square Statistic, n-1 )

Tail Area Right of Chi-Square Statistic =

CHIDIST( 69.09766, 49 ) = 3.07

So area under the curve outside the Chi-Square Statistic = 3.07%

The P Value = 0.0307

The Red Area Outside the Chi-Square Statistic
(Is Smaller Than Alpha) in the Outer Right Tail
So We Can State With 95% Certainty That
the Population Variance Have Moved to the
Right (Increased)
Click on the Image To See a Larger Version

Step 5) Analyze Using the Chi-Square Statistic Rule: If the P Value (P Value = 0.0307), the area under the curve outside the Chi-Square Statistic, is less than α (α = 0.05), the population variance has moved in the direction of Sample Standard Deviation.

We can see that the red area fits inside of the yellow area on the outer right tail. In this case, the area outside the Chi-Statistic (3.07%) is less than Alpha (5%), we can state with 95% certainty that the population variance has increased.

Now the Internet marketing manager needs to determine the underlying reason why customer spending has become less focused.

Conclusion - The Chi-Square Variance Test Tells Whether Something Has Changed Your Customers' Buying Habits

Incidentally, all of the Chi-Square Probability Density Function graphs in this article had 49 Degrees of Freedom. Degrees of Freedom is derived from sample size and equals n-1 (50 - 1 = 49 Degrees of Freedom).

Here are links to other training videos of how to create interactive graphs in Excel of some of the other major statistical distributions:
How to Graph the Normal Distribution's Probability Density Function in Excel

How To Graph the Normal Distribution's Cumulative Distribution Function in Excel

How To Graph the Students t Distributions' Probability Density Function in Excel

How To Graph the Chi-Square Distribution's Probability Density Function in Excel

How To Graph the Weibull Distribution's PDF and CDF - in Excel

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