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What you’ll learn about in this chapter

• Why variability is valuable as a descriptive tool• How to compute the range, standard deviation, and variance• How the standard deviation and variance are alike, and how

they are different• Using the Analysis ToolPak to compute the range, standard

deviation, and variance

WHY UNDERSTANDINGVARIABILITY IS IMPORTANT

In Chapter 2, you learned about different types of averages, whatthey mean, how they are computed, and when to use them. Butwhen it comes to descriptive statistics and describing the character-istics of a distribution, averages are only half the story. The otherhalf is measures of variability.

In the most simple of terms, vvaarriiaabbiilliittyy reflects how scores differfrom one another. For example, the following set of scores showssome variability:

7, 6, 3, 3, 1

3 VViivvee llaa DDiifffféérreennccee

Understanding Variability

Difficulty Scale ☺☺☺☺ (moderately easy, but not a cinch)

How much Excel? (a ton)

66

The following set of scores has the same mean (4) and has lessvariability than the previous set:

3, 4, 4, 5, 4

The next set has no variability at all—the scores do not differfrom one another—but it also has the same mean as the other twosets we just showed you.

4, 4, 4, 4, 4

Variability (also called spread or dispersion) can be thought of asa measure of how different scores are from one another. It’s evenmore accurate (and maybe even easier) to think of variability ashow different scores are from one particular score. And what “score”do you think that might be? Well, instead of comparing each scoreto every other score in a distribution, the one score that could beused as a comparison is—that’s right—the mean. So, variabilitybecomes a measure of how much each score in a group of scores dif-fers from the mean. More about this in a moment.

Remember what you already know about computing averages—that an average (whether it is the mean, the median, or the mode)is a representation of a set of scores. Now, add your new knowledgeabout variability—that it reflects how different scores are from oneanother. Each is an important descriptive statistic. Together, thesetwo (average and variability) can be used to describe the character-istics of a distribution and show how distributions differ from oneanother.

Three measures of variability are commonly used to reflect thedegree of variability, spread, or dispersion in a group of scores.

These are the range, the standard deviation, and the variance.Let’s take a closer look at each one and how each one is used.

COMPUTING THE RANGE

The range is the most general measure of variability. It gives you anidea of how far apart scores are from one another. The rraannggee is com-puted simply by subtracting the lowest score in a distribution fromthe highest score in the distribution.

In general, the formula for the range is

r = h – l, (3.1)

Chapter 3 ♦ Vive la Différence 67

where

r is the range,

h is the highest score in the data set, and

l is the lowest score in the data set.

Take the following set of scores, for example (shown here indescending order):

98, 86, 77, 56, 48

In this example, 98 – 48 = 50. The range is 50.

There really are two kinds of ranges. One is the exclusive range, whichis the highest score minus the lowest score (or h – l) and the one we justdefined. The second kind of range is the inclusive range, which is thehighest score minus the lowest score plus 1 (or h – l + 1). You most com-monly see the exclusive range in research articles, but the inclusiverange is also used on occasion if the researcher prefers it.

The range is used almost exclusively to get a very general esti-mate of how wide or different scores are from one another—that is,the range shows how much spread there is from the lowest to thehighest point in a distribution.

So, although the range is fine as a general indicator of variability,it should not be used to reach any conclusions regarding how indi-vidual scores differ from one another. Remember, the range usesonly two scores—a poor reflection of what’s really happening.

And as far as Excel is concerned, there is no function for comput-ing the range. Rather, you can create a simple formula that subtractsone value from the other and adds 1 (to compute an inclusiverange) or doesn’t add anything for the exclusive range.

COMPUTING THE STANDARD DEVIATION

Now we get to the most frequently used measure of variability, thestandard deviation. Just think about what the term implies; it’s adeviation from something (guess what?) that is standard. Actually,the ssttaannddaarrdd ddeevviiaattiioonn (abbreviated as s or SD) represents the aver-age amount of variability in a set of scores. In practical terms, it’s the

68 Part II ♦ ∑igma Freud and Descriptive Statistics

average distance from the mean. The larger the standard deviation,the larger the average distance each data point is from the mean ofthe distribution.

So, what’s the logic behind computing the standard deviation? Yourinitial thoughts may be to compute the mean of a set of scores andthen subtract each individual score from the mean. Then, computethe average of that distance. That’s a good idea—you’ll end up withthe average distance of each score from the mean. But it won’t work(see if you know why even though we’ll show you why in a moment).

First, here’s the formula for computing the standard deviation:

(3.2)

where

s is the standard deviation;

∑ is sigma, which tells you to find the sum of what follows;

X is each individual score;

X–

is the mean of all the scores; and

n is the sample size.

This formula finds the difference between each individual scoreand the mean (X – X

–), squares each difference, and sums them all

together. Then, it divides the sum by the size of the sample (minus 1)and takes the square root of the result. As you can see, and as wementioned earlier, the standard deviation is an average deviationfrom the mean.

Here are the data we’ll use in the following step-by-step explana-tion of how to compute the standard deviation.

5, 8, 5, 4, 6, 7, 8, 8, 3, 6

1. List each score. It doesn’t matter whether the scores are in any par-ticular order.

2. Compute the mean of the group.

3. Subtract the mean from each score.

Here’s what we’ve done so far, where X – X–

represents the differencebetween the actual score and the mean of all the scores, which is 6.

s=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP ðX − �XÞ2

n− 1

s

,


X X– X – X–

8 6 8 – 6 = +2

8 6 8 – 6 = +2

8 6 8 – 6 = +2

7 6 7 – 6 = +1

6 6 6 – 6 = 0

6 6 6 – 6 = 0

5 6 5 – 6 = –1

5 6 5 – 6 = –1

4 6 4 – 6 = –2

3 6 3 – 6 = –3

4. Square each individual difference. The result is the column marked(X – X– )

2.

X (X – X–) (X – X–)2

8 +2 48 +2 48 +2 47 +1 16 0 06 0 05 –1 15 –1 14 –2 43 –3 9Sum 0 28

5. Sum all the squared deviations about the mean. As you can seeabove, the total is 28.

6. Divide the sum by n – 1, or 10 – 1 = 9, so then 28/9 = 3.11.

7. Compute the square root of 3.11, which is 1.76 (after rounding).That is the standard deviation for this set of 10 scores.

And Now . . . Using Excel’s STDEV Function

To compute the standard deviation of a set of numbers usingExcel, follow these steps:

1. Enter the individual scores into one column in a worksheetsuch as you see in Figure 3.1.


2. Select the cell into which you want to enter the SSTTDDEEVV func-tion. In this example, we are going to compute the standarddeviation in Cell B12.

3. Now click on Cell B12 and type the STDEV function as follows:=STDEV(B2:B11)and press the Enter key,oruse the Formulas � Insert Function menu option and the“Inserting a Function” technique we talked about on pages 25through 29 in Chapter 1 to enter the STDEV function in Cell B12.

4. As you can see in Figure 3.2, the standard deviation wascomputed and the value (1.76, just as we did manually ear-lier) returned to Cell B12. Notice that in the formula bar inFigure 3.2, you can see the STDEV function fully expressed.


Figure 3.1 Data for the STDEV Function

Figure 3.2 The Computation of the Standard Deviation Usingthe STDEV Function

More Excel

What Function Loves Ya, Baby

Now, here’s a surprise. When you go to select the STDEV function fromthe Formulas � Insert Function options, you see that there are actuallyseveral functions that can compute the standard deviation. There aretwo especially important ones for us. One is named STDEV (the one weused in the example above), and the second is named STDEVP. Theyboth compute the standard deviation of a set of scores, but what’s thedifference?

The one that we used, STDEV, computes the standard deviation fora set of scores from a sample. STDEVP computes the standard devia-tion for a set of scores that is an entire population. What’s the differ-ence in the two values, and why use one rather than the other? Moreabout this later, but for now, because STDEV is based on only a portionof the entire population (the definition of a sample, right?), it’s morelikely (we’re talking probability here) to overestimate the true, real,honest-to-Pete value of the standard deviation when computed for theentire population.

When to use what? Use STDEV—you’ll almost always be correct.But if a group of scores is defined as a population, then STDEVP isthe function that loves ya, baby.

What we now know from these results is that each score in thisdistribution differs from the mean by an average of 1.76 points.

Let’s take a short step back and examine some of the operationsin the standard deviation formula. They’re important to review andwill increase your understanding of what the standard deviation is.

First, why didn’t we just add up the deviations from the mean?Because the sum of the deviations from the mean is always equal to 0.Try it by summing the deviations (2 + 2 + 2 + 1 + 0 + 0 – 1 – 1 – 2 – 3).In fact, that’s the best way to check if you computed the mean correctly.

There’s another type of deviation that you may read about, and youshould know what it means. The mean deviation (also called the meanabsolute deviation) is the average of the absolute value of the devia-tions from the mean. You already know that the sum of the deviationsfrom the mean must equal 0 (otherwise, the mean probably has beencomputed incorrectly). Instead, let’s take the absolute value of eachdeviation (which is the value regardless of the sign). Sum them togetherand divide by the number of data points, and you have the mean devi-ation. (Note: The absolute value of a number is usually represented asthat number with a vertical line on each side of it, such as |5|. Forexample, the absolute value of –6, or |–6|, is 6.)



Second, why do we square the deviations? Because we want to getrid of the negative sign so that when we do eventually sum them,they don’t add up to 0.

And finally, why do we eventually end up taking the square rootof the entire value in Step 7? Because we want to return to the sameunits with which we originally started. We squared the deviationsfrom the mean in Step 4 (to get rid of negative values) and then tookthe square root of their total in Step 7. Pretty tidy.

Why n – 1? What’s Wrong With Just n?

You might have guessed why we square the deviations about themean and why we go back and take the square root of their sum. Buthow about subtracting the value of 1 from the denominator of theformula? Why do we divide by n – 1 rather than just plain ol’ n?Good question.

The answer is that s (the standard deviation) is an estimate of thepopulation standard deviation and is an uunnbbiiaasseedd eessttiimmaattee at that, butonly when we subtract 1 from n. By subtracting 1 from the denomina-tor, we artificially force the standard deviation to be larger than it wouldbe otherwise. Why would we want to do that? Because, as good scien-tists, we are conservative, and that is exactly what we are doing by usingSTDEV rather than the STDEVP function in the earlier example.

In fact, you can see that if you look at Figure 3.3, where we usedboth functions to compute the standard deviation. See how theunbiased one (1.76) is larger than the biased one (1.67)? That’sbecause the one based on the sample (the unbiased one) intention-ally overestimates the value.

Figure 3.3 A Comparison of the STDEV and STDEVP Functions

Being conservative means that if we have to err, we will do so onthe side of overestimating what the standard deviation of the popu-lation is. Dividing by a smaller denominator lets us do so. Thus,instead of dividing by 10, we divide by 9. Or, instead of dividing by100, we divide by 99.

Biased estimates are appropriate if your intent is only to describe thecharacteristics of the sample. But if you intend to use the sample as anestimate of a population parameter, then the unbiased statistic is bestto calculate.

Take a look in the following table and see what happens as the sizeof the sample gets larger (and moves closer to the population in size).The n – 1 adjustment has far less of an impact on the differencebetween the biased and the unbiased estimates of the standard devia-tion (the bold column in the table). All other things being equal, then,the larger the size of the sample, the less of a difference there isbetween the biased and the unbiased estimates of the standard devia-tion. Check out the following table, and you’ll see what we mean.

Biased Unbiased Estimate of Estimate of Difference

Value of the Population the Population Between Numerator in Standard Standard Biased and

Standard Deviation Deviation Unbiased Sample Size Deviation (dividing by n) (dividing by n – 1) Estimates

10 500 7.07 7.45 .38

100 500 2.24 2.25 .01

1,000 500 0.7071 0.7075 .0004

The moral of the story? When you compute the standard devia-tion for a sample, which is an estimate of the population, the closerto the size of the population the sample is, the more accurate theestimate will be.

What’s the Big Deal?

The computation of the standard deviation is very straightfor-ward. But what does it mean? As a measure of variability, all it tellsus is how much each score in a set of scores, on the average, variesfrom the mean. But it has some very practical applications, as youwill find out in Chapter 4. Just to whet your appetite, consider this:


The standard deviation can be used to help us compare scoresfrom different distributions, even when the means and standarddeviations are different.

Amazing! This, as you will see, can be very cool.

THINGS TO REMEMBER

The standard deviation is computed as the average distance fromthe mean. So, you will need to first compute the mean as a measureof central tendency. Don’t fool around with the median or the modein trying to compute the standard deviation. The larger the standarddeviation, the more spread out the values are, and the more differ-ent they are from one another.

Just like the mean, the standard deviation is sensitive to extremescores. When you are computing the standard deviation of a sample andyou have extreme scores, note that somewhere in your written report.

If s = 0, there is absolutely no variability in the set of scores, andthey are essentially identical in value. This will rarely happen.

COMPUTING THE VARIANCE

Here comes another measure of variability and a nice surprise. Ifyou know the standard deviation of a set of scores and you cansquare a number, you can easily compute the variance of that sameset of scores. This third measure of variability, the vvaarriiaannccee,, issimply the standard deviation squared. In other words, it’s the sameformula you saw earlier, without the square root bracket, like theone shown in Formula 3.3:

(3.3)

If you take the standard deviation and never complete the laststep (taking the square root), you have the variance. In other words,s2 = s × s, or the variance equals the standard deviation times itself(or squared). In our earlier example, where the standard deviationwas equal to 1.76, the variance is equal to 1.762 or 3.10.

s2 =P ðX − �XÞ2

n− 1:



And Now . . . Using Excel’s VAR Function

To compute the variance of a set of numbers using Excel, followthese steps:

1. Enter the individual scores into one column in a worksheet.We’re going to use the same scores as those shown in Figure 3.1.

2. Select the cell into which you want to enter the VVAARR function. Inthis example, we are going to compute the variance in Cell B12.

3. Now click on Cell B12 and type the VAR function as follows:=VAR(B2:B11)and press the Enter key,oruse the Formula � Insert Function menu option and the“Inserting a Function” technique we talked about on pages 25through 29 in Chapter 1 to enter the VAR function in Cell B12.

4. As you can see in Figure 3.4, the variance was computed andthe value returned to Cell B12. Notice that in the formula barin Figure 3.4, you can see the VAR function fully expressedand the value computed as 3.10, just as we did manually ear-lier in the chapter. You will notice that we computed this valueto be 3.10, but Figure 3.4 shows it to be 3.11. That’s a func-tion of the internal formula that Excel uses to compute thevariance and nothing more.

Figure 3.4 Using the VAR Function to Compute the Variance

More Excel

STDEV Is to STDEVP as VAR Is to VARPSounds like an acronym love fest, right? Nope—just as there is a corre-sponding function for computing the standard deviation of a population(STDEVP), so there is for the variance as well, and it is named VARP. InFigure 3.5, you can see how when each is computed, like the unbiasednature of STDEV or VAR, they are larger than their STDEVP and VARP cousins.


Figure 3.5 Comparing the VAR and VARP Functions

You are not likely to see the variance mentioned by itself in ajournal article or see it used as a descriptive statistic. This is becausethe variance is a difficult number to interpret and to apply to a setof data. After all, it is based on squared deviation scores.

But the variance is important because it is used both as a conceptand as a practical measure of variability in many statistical formulasand techniques. You will learn about these later in Statistics for PeopleWho (Think They) Hate Statistics . . . Excel 2007 Edition.

The Standard Deviation Versus the Variance

How are standard deviation and the variance the same, and howare they different?

Well, they are both measures of variability, dispersion, or spread.The formulas used to compute them are very similar. You see themall over the place in the “Results” sections of journals.

They are also quite different.First, and most important, the standard deviation (because we

take the square root of the average summed squared deviation) is

stated in the original units from which it was derived. The varianceis stated in units that are squared (the square root is never taken).

What does this mean? Let’s say that we need to know the variabilityof a group of production workers assembling circuit boards. Let’s say thatthey average 8.6 boards per hour, and the standard deviation is 1.59. Thevalue 1.59 means that the difference in the average number of boardsassembled per hour is about 1.59 circuit boards from the mean.

Let’s look at an interpretation of the variance, which is 1.592, or2.53. This would be interpreted as meaning that the average differ-ence between the workers is about 2.53 circuit boards squared fromthe mean. Which of these two makes more sense?

USING THE AMAZING ANALYSIS TOOLPAK (AGAIN!)

Guess what? We already did this! See pages 57 and 58 in Chapter 2where the ToolPak does a descriptive analysis including the standarddeviation, the variance, and the range. Now, that was easy, wasn’t it?

Summary

Measures of variability help us understand even more fully what a dis-tribution of data points looks like. Along with a measure of central ten-dency, we can use these values to distinguish distributions from oneanother and effectively describe what a collection of test scores, heights,or measures of personality looks like. Now that we can think and talkabout distributions, let’s look at ways we can look at them.

Time to Practice

1. Why is the range the most convenient measure of dispersion, yet themost imprecise measure of variability? When would you use the range?

2. For the following set of scores, compute the range, the unbiased and thebiased standard deviation, and the variance. Do the exercise by hand.

31, 42, 35, 55, 54, 34, 25, 44, 35

3. Why is the unbiased estimate a larger value than the biased estimate?

4. This practice problem uses the data contained in the file named Chapter 3Data Set 1 in Appendix C. There are two variables in this data set.

Variable Definition

Height height in inchesWeight weight in pounds


Using Excel, compute all of the measures of variability you can forheight and weight. Does Excel compute a biased or an unbiased esti-mate? How do you know?

5. You’re one of the outstanding young strategists for Western Airlinesand are really curious to examine the number of passengers flying ina morning–evening flight comparison out of the Kansas City;Philadelphia; and Providence, Rhode Island, hubs for the last 2 daysof last week. Here are the data—compute the descriptive statisticsyou need that you will use in tomorrow’s presentation to your bossand provide a few sentences of summary. Be on time.

Thursday Friday Thursday Friday Thursday Friday

Morning To To To To To ToFlights Kansas Kansas Philadelphia Philadelphia Providence Providence

City City

Number of 258 251 303 312 166 176passengers

Evening To To To To To To Flights Kansas Kansas Philadelphia Philadelphia Providence Providence

City City

Number of 312 331 321 331 210 274passengers

Answers to Practice Questions

1. The range is the most convenient measure of dispersion because itrequires you only to subtract one number (the lowest value) fromanother number (the highest value). It’s imprecise because it does nottake into account the values that fall between the highest and the low-est values in a distribution. Use the range when you want a very gross(and not very precise) estimate of the variability in a distribution.

2. The range is 30. The unbiased sample standard deviation equals10.19. The biased estimate equals 9.60. The difference is due todividing by a sample size of 8 (for the unbiased estimate) as com-pared to a sample size of 9 (for the biased estimate). The unbiasedestimate of the variance is 103.78, and the biased estimate is 92.25.

3. The unbiased estimate should be larger than the biased estimatebecause we are intentionally being conservative and overestimating thesize of the population standard deviation. Arithmetically, for the biasedestimate, the denominator is larger and hence the value is smaller.

4. Figure 3.6 shows you the data and the computed standard deviationand variance. In both cases, we used the unbiased functions STDEVand VAR to compute the values. The only way to know whether the


functions compute a biased or an unbiased estimate is if you knowwhich function was selected to compute these values. After the fact,you can, of course, just look at the formula bar and examine thecell’s contents.


Figure 3.6 Computing the Standard Deviation and the VarianceUsing STDEV and VAR

5. First, the facts . . . The average number of folks flying on a morningflight is 244 (actually 244.33), and the average number flying on anevening flight is 296 (actually 296.5).The standard deviation for morning flights is 61.74 and for evening

flights is 47.35.You can compute just about any measures of central tendency and

variation and make some sense of them. For example,Just on the surface (and there are a lot more descriptive things we

could compute that might be of interest, such as a city by time-of-daycomparison), more people fly in the evening than the morning (whichmight mean more one-way tickets are purchased, etc.) and the numberof people tends to be more consistent (a lower standard deviation) whenit comes to evening flights.

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