encyclopedia of research design central tendency, measures of · primary measures of central...

Encyclopedia of Research Design

Central Tendency, Measures of

Contributors: Carol A. CarmanEditors: Neil J. SalkindBook Title: Encyclopedia of Research DesignChapter Title: "Central Tendency, Measures of"Pub. Date: 2010Access Date: December 10, 2014Publishing Company: SAGE Publications, Inc.City: Thousand OaksPrint ISBN: 9781412961271Online ISBN: 9781412961288DOI: http://dx.doi.org/10.4135/9781412961288.n46Print pages: 138-143

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of 14 Encyclopedia of Research Design: CentralTendency, Measures of

http://dx.doi.org/10.4135/9781412961288.n46One of the most common statistical analyses used in descriptive statistics is a processto determine where the average of a set of values falls. There are multiple ways todetermine the middle of a group of numbers, and the method used to find the averagewill determine what information is known and how that average should be interpreted.Depending on the data one has, some methods for finding the average may be moreappropriate than others.

The average describes the typical or most common number in a group of numbers. Itis the one value that best represents the entire group of values. Averages are used inmost statistical analyses, and even in everyday life. If one wanted to find the typicalhouse price, family size, or score on a test, some form of average would be computedeach time. In fact, one would compute a different type of average for each of thosethree examples.

Researchers use different ways to calculate the average, based on the types ofnumbers they are examining. Some numbers, measured on the nominal level ofmeasurement, are not appropriate to do some types of averaging. For example, if onewere examining the variable of types of vegetables, and the labels of the levels werecucumbers, zucchini, carrots, and turnips, if one performed some types of average,one may find that the average vegetable was 3/4 carrot and 1/4 turnip, which makesno sense at all. Similarly, using some types of average on interval-level continuousvariables may result in an average that is very imprecise and not very representative ofthe sample one is using. When the three main methods for examining the average arecollectively discussed, they are referred to as measures of central tendency. The threeprimary measures of central tendency commonly used by researchers are the mean,the median, and the mode.

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Mean

The mean is the most commonly used (and misused) measure of central tendency. Themean is defined as the sum of all the scores in the sample, divided by the number of

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scores in the sample. This type of mean is also referred to as the arithmetic mean, todistinguish it from other types of means, such as the geometric mean or the harmonicmean. Several common symbols or statistical notations are used to represent the mean,including

, which is read as x-bar (the mean of the sample), and µ, which is read as mu (the meanof the population). Some research articles also use an italicized uppercase letter M toindicate the sample mean.

Much as different symbols are used to represent the mean, different formulas are usedto calculate the mean. The difference between the two most common formulas is foundonly in the symbols used, as the formula for calculating the mean of a sample uses thesymbols appropriate for a sample, and the other formula is used to calculate the meanof a population, and as such uses the symbols that refer to a population.

For calculating the mean of a sample, use

For calculating the mean of a population, use

Calculating the mean is a very simple process. For example, if a student had turnedin five homework assignments that were worth 10 points each, the student's scoreson those assignments might have been 10, 8, 5, 7, and 9. To calculate the student'smean score on the homework assignments, first one would add the values of all thehomework assignments:

Then one would divide the sum by the total number of assignments (which was 5):



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For this example, 7.8 would be the student's mean score for the five homeworkassignments. That indicates that, on average, the student scored a 7.8 on eachhomework assignment.

Other Definitions

Other definitions and explanations can also be used when interpreting the mean. Onecommon description is that the mean is like a balance point. The mean is located at thecenter of the values of the sample or population. If one were to line up the values ofeach score on a number line, the mean would fall at the exact point where the valuesare equal on each side; the mean is the point closest to the squared distances of all thescores in the distribution.

Another way of thinking about the mean is as the amount per individual, or how mucheach individual would receive if one were to divide the total amount equally. If Stevehas a total of $50, and if he were to divide it equally among his four friends, each friendwould receive $50/4 = $12.50. Therefore, $12.50 would be the mean.

The mean has several important properties that are found only with this measure ofcentral tendency. If one were to change a score in the sample from one value to anothervalue, the calculated value of the mean would change. The value of the mean wouldchange because the value of the sum of all the scores would change, thus changingthe numerator. For example, given the scores in the earlier homework assignmentexample (10, 8, 5, 7, and 9), the student previously scored a mean of 7.8. However, ifone were to change the value of the score of 9 to a 4 instead, the value of the meanwould change:

By changing the value of one number in the sample, the value of the mean was loweredby one point. Any change in the value of a score will result in a change in the value ofthe mean.



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If one were to remove or add a number to the sample, the value of the mean wouldalso change, as then there would be fewer (or greater) numbers in the sample, thuschanging the numerator and the denominator. For example, if one were to calculate themean of only the first four homework assignments (thereby removing a number from thesample),

[p. 139 ↓ ]

If one were to include a sixth homework assignment (adding a number to the sample)on which the student scored an 8,

Either way, whether one adds or removes a number from the sample, the mean willalmost always change in value. The only instance in which the mean will not change isif the number that is added or removed is exactly equal to the mean. For example, if thescore on the sixth homework assignment had been 7.8,

If one were to add (or subtract) a constant to each score in the sample, the mean willincrease (or decrease) by the same constant value. So if the professor added threepoints to the score on every homework assignment,

The mean homework assignment score increased by 3 points. Similarly, if the professortook away two points from each original homework assignment score,



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The mean homework assignment score decreased by two points. The same type ofsituation will occur with multiplication and division. If one multiplies (or divides) everyscore by the same number, the mean will also be multiplied (or divided) by that number.Multiplying the five original homework scores by four,

results in the mean being multiplied by four as well. Dividing each score by 3,

will result in the mean being divided by three as well.

Weighted Mean

Occasionally, one will need to calculate the mean of two or more groups, each of whichhas its own mean. In order to get the overall mean (sometimes called the grand meanor weighted mean), one will need to use a slightly different formula from the one used tocalculate the mean for only one group:

To calculate the weighted mean, one will divide the sum of all the scores in every groupby the number of scores in every group. For example, if Carla taught three classes, andshe gave each class a test, the first class of 25 students might have a mean of 75, thesecond class of 20 students might have a mean of 85, and the third class of 30 studentsmight have a mean of 70. To calculate the weighted mean, Carla would first calculatethe summed scores for each class, then add the summed scores together, then divideby the total number of students in all three classes. To find the summed scores, Carlawill need to rework the formula for the sample mean to find the summed scores instead:



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With this reworked formula, find each summed score first:

[p. 140 ↓ ]

Next, add the summed scores together:

Then add the number of students per class together:

Finally, divide the total summed scores by the total number of students:

This is the weighted mean for the test scores of the three classes taught by Carla.

Median

The median is the second measure of central tendency. It is defined as the score thatcuts the distribution exactly in half. Much as the mean can be described as the balancepoint, where the values on each side are identical, the median is the point where thenumber of scores on each side is equal. As such, the median is influenced more by thenumber of scores in the distribution than by the values of the scores in the distribution.The median is also the same as the 50th percentile of any distribution. Generally themedian is not abbreviated or symbolized, but occasionally Mdn is used.

The median is simple to identify. The method used to calculate the median is the samefor both samples and populations. It requires only two steps to calculate the median. Inthe first step, order the numbers in the sample from lowest to highest. So if one wereto use the homework scores from the mean example, 10, 8, 5, 7, and 9, one would first



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order them 5, 7, 8, 9, 10. In the second step, find the middle score. In this case, there isan odd number of scores, and the score in the middle is 8.

Notice that the median that was calculated is not the same as the mean for the samesample of homework assignments. This is because of the different ways in which thosetwo measures of central tendency are calculated.

It is simple to find the middle when there are an odd number of scores, but it is a bitmore complex when the sample has an even number of scores. For example, whenthere were six homework scores (10, 8, 5, 7, 9, and 8), one would still line up thehomework scores from lowest to highest, then find the middle:

In this example, the median falls between two identical scores, so one can still say thatthe median is 8. If the two middle numbers were different, one would find the middlenumber between the two numbers. For example, if one increased one of the student'shomework scores from an 8 to a 9,

In this case, the middle falls halfway between 8 and 9, at a score of 8.5.

Statisticians disagree over the correct method for calculating the median when thedistribution has multiple repeated scores in the center of the distribution. Somestatisticians use the methods described above to find the median, whereas othersbelieve the scores in the middle need to be reduced to fractions to find the exactmidpoint of the distribution. So in a distribution with the following scores,

some statisticians would say the median is 5, whereas others (using the fractionmethod) would report the median as 4.7.



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Mode

The mode is the last measure of central tendency. It is the value that occurs mostfrequently. It is the simplest and least precise measure of central tendency. Generally,in writing about the mode, scholars label it simply mode, although some books orpapers use Mo as an abbreviation. The method for finding the mode is the same forboth samples and populations. Although there are several ways one could find themode, a simple method is to list each score that appears in the [p. 141 ↓ ] sample. Thescore that appears the most often is the mode. For example, given the following sampleof numbers,

one could arrange them in numerical order:

Once the numbers are arranged, it becomes apparent that the most frequentlyappearing number is 4:

Thus 4 is the mode of that sample of numbers. Unlike the mean and the median, it ispossible to have more than one mode. If one were to add two threes to the sample,

then both 3 and 4 would be the most commonly occurring number, and the mode ofthe sample would be 3 and 4. The term used to describe a sample with two modesis bimodal. If there are more than two modes in a sample, one says the sample ismultimodal.



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When to Use Each Measure

Because each measure of central tendency is calculated with a different method, eachmeasure is different in its precision of measuring the middle, as well as which numbersit is best suited for.

Mean

The mean is often used as the default measure of central tendency. As most peopleunderstand the concept of average, they tend to use the mean whenever a measure ofcentral tendency is needed, including times when it is not appropriate to use the mean.Many statisticians would argue that the mean should not be calculated for numbers thatare measured at the nominal or ordinal levels of measurement, due to difficulty in theinterpretation of the results. The mean is also used in other statistical analyses, suchas calculations of standard deviation. If the numbers in the sample are fairly normallydistributed, and there is no specific reason that one would want to use a differentmeasure of central tendency, then the mean should be the best measure to use.

Median

There are several reasons to use the median instead of the mean. Many statisticiansbelieve that it is inappropriate to use the mean to measure central tendency if thedistribution was measured at the ordinal level. Because variables measured at theordinal level contain information about direction but not distance, and because the meanis measured in terms of distance, using the mean to calculate central tendency wouldprovide information that is difficult to interpret.

Another occasion to use the median is when the distribution contains an outlier. Anoutlier is a value that is very different from the other values. Outliers tend to be locatedat the far extreme of the distribution, either high or low. As the mean is so sensitiveto the value of the scores, using the mean as a measure of central tendency in adistribution with an outlier would result in a nonrepre-sentative score. For example,



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looking again at the five homework assignment scores, if one were to replace the ninewith a score of 30,

By replacing just one value with an outlier, the newly calculated mean is not a goodrepresentation of the average values of our distribution. The same would occur if onereplaced the score of 9 with a very low number:

Since the mean is so sensitive to outliers, it is best to use the median for calculatingcentral tendency. Examining the previous example, but using the median,

[p. 142 ↓ ]

The middle number is 7, therefore the median is 7, a much more representative numberthan the mean of that sample, 5.2.

If the numbers in the distribution were measured on an item that had an open-endedoption, one should use the median as the measure of central tendency. For example, aquestion that asks for demographic information such as age or salary may include eithera lower or upper category that is open ended:

Number of Cars Owned Frequency

0 4

1 10

2 16

3 or more 3



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The last answer option is open ended because an individual with three cars wouldbe in the same category as someone who owned 50 cars. As such, it is impossibleto accurately calculate the mean number of cars owned. However, it is possible tocalculate the median response. For the above example, the median would be 2 carsowned.

A final condition in which one should use the median instead of the mean is when onehas incomplete information. If one were collecting survey information, and some of theparticipants refused or forgot to answer a question, one would have responses fromsome participants but not others. It would not be possible to calculate the mean in thisinstance, as one is missing important information that could change the mean thatwould be calculated if the missing information were known.

Mode

The mode is well suited to be used to measure the central tendency of variablesmeasured at the nominal level. Because variables measured at the nominal levelare given labels, then any number assigned to these variables does not measurequantity. As such, it would be inappropriate to use the mean or the median withvariables measured at this level, as both of those measures of central tendency requirecalculations involving quantity. The mode should also be used when finding the middleof distribution of a discrete variable. As these variables exist only in whole numbers,using other methods of central tendency may result in fractions of numbers, making itdifficult to interpret the results. A common example of this is the saying, “The averagefamily has a mom, a dad, and 2.5 kids.” People are discrete variables, and as such,they should never be measured in such a way as to obtain decimal results. The modecan also be used to provide additional information, along with other calculations ofcentral tendency. Information about the location of the mode compared with the meancan help determine whether the distribution they are both calculated from is skewed.

Carol A.Carman

http://dx.doi.org/10.4135/9781412961288.n46See also



http://dx.doi.org/10.4135/9781412961288.n46

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• Descriptive Statistics• Levels of Measurement• Mean• Median• Mode• “On the Theory of Scales of Measurement”• Results Section• Sensitivity• Standard Deviation• Variability, Measure of

Further Readings

Coladarci, T., Cobb, C. D., Minium, E. W., & Clarke, R. C. (2004). Fundamentals ofstatistical reasoning in education . Hoboken, NJ: Wiley.

Gravetter, F. J., & Wallnau, L. B. (2004). Statistics for the behavioral sciences (6th ed.).Belmont, CA: Thomson Wadsworth.

Salkind, N. J. (2008). Statistics for people who (think they) hate statistics . ThousandOaks, CA: Sage.

Thorndike, R. M. (2005). Measurement and evaluation in psychology and education .(7th ed.). Upper Saddle River, NJ: Pearson Education.



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