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Applied Statistics Handbook
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Measures of Central Tendency

Mode: The most frequently occurring score.  A distribution of scores can be unimodal (one score occurred most frequently), bimodal (two scores tied for most frequently occurring), or multimodal. In the table below the mode is 32.  If there were also two scores with the value of 60, we would have a bimodal distribution (32 and 60).

Median: The point on a rank ordered list of scores below which 50% of the scores fall.  It is especially useful as a measure of central tendency when there are very extreme scores in the distribution, such as would be the case if we had someone in the age distribution provided below who was 120.  If the number of scores is odd, the median is the score located in the position represented by (n+1)/2. In the table below the median is located in the 4th position (7+1)/2 and would be reported as a median of 42.  If the number of scores are even, the median is the average of the two middle scores. As an example, if we dropped the last score (65) in the above table, the median would be represented by the average of the 3rd (6/2) and 4th score, or 37 (32+42)/2. Always remember to order the scores from low to high before determining the median.

 

Variable

Age

Also known as X

 

 

 

24

 

 

 

 

32

Mode

 

 

 

32

 

 

 

 

42

Median

Xi

Each score

 

55

 

 

 

 

60

 

 

 

 

65

 

 

 

 

 

 

 

 

n=

7

Number of scores (or cases)

 

 

310

Sum of scores

 

 

44.29

Mean

 

 

Mean: The sum of the scores () is divided by the number of scores (n) to compute an arithmetic average of the scores in the distribution.  The mean is the most often used measure of central tendency.   It has two properties: 1) the sum of the deviations of the individual scores (Xi) from the mean is zero, 2) the sum of squared deviations from the mean is smaller than what can be obtained from any other value created to represent the central tendency of the distribution. In the above table the mean age is 44.29 (310/7).

Weighted Mean: When two or more means are combined to develop an aggregate mean, the influence of each mean must be weighted by the number of cases in its subgroup.

 

 

Example

 

 

Wrong Method:          

Correct Method:         

 

Measures of Variation

Range:  The difference between the highest and lowest score (high-low).  It describes the span of scores but cannot be compared to distributions with a different number of observations.  In the table below, the range is 41 (65-24).

Variance:  The average of the squared deviations between the individual scores and the mean.  The larger the variance the more variability there is among the scores.   When comparing two samples with the same unit of measurement (age), the variances are comparable even though the sample sizes may be different.  Generally, however, smaller samples have greater variability among the scores than larger samples.  The sample variance for the data in the table below is 251.57.   The formula is almost the same for estimating population variance.  See formula in Appendix.

Standard deviation: The square root of variance. It provides a representation of the variation among scores that is directly comparable to the raw scores.  The sample standard deviation in the following table is 15.86 years.

 

Variable

Age

44.29

 

24

-20.29

411.68

 

 

32

-12.29

151.04

 

 

32

-12.29

151.04

squared deviations

 

42

-2.29

5.24

 

 

55

10.71

114.70

 

 

60

15.71

246.80

 

 

65

20.71

428.90

 

 

 

 

 

 

n=

7

 

1509.43

 

 

251.57

sample variance

 

 

15.86

sample standard deviation

 

Software Output Example


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