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Chapter 4 DESCRIPTIVE STATISTICS: MEASURES OF CENTRAL TENDENCY AND VARIABILITY Understanding Statistics for International Social Work and Other Behavioral Sciences Serge Lee, Maria C Silveira Nunes Dinis, Lois Lowe, Kelly Anders (2015. Oxford University Press MEASURES OF CENTRAL TENDENCY 2 It is used to determine how values, particularly interval and ratio data, in a given distribution of scores are clustered and identify the key locations in the set. For example, X = 5, 1, 4, 5, 3, 2, and 5. Central locations can be found by using the mean, median, and mode The mean X . The mean can be computed by: • X= X n • X = Sample mean • X = Sum of the variable (x) Sample mean is the arithmetic average. It is also called simple statistic. • X= 5+1+4+5+4+2 7 • X = 3.57 MEASURES OF CENTRAL TENDENCY CONTINUED 3 In addition to the sample mean, trimmed mean can also be computed. Typically, 15-20% is trimmed from the data set. For example, X = 1, 4, 5, 3, 5, 8, 4, 5 and 9. Assume 20% is trimmed. Compute the trimmed mean by: Multiply the selection proportion (in decimal) by the sample size (.20 x 9 = 1.80) Arrange the scores into an array, 1, 3, 4, 4, 5, 5, 5, 8, 9 Trim first two (1, 3) and last two (8, 9) from the set Take the average of the remaining scores, = 4.6. Without trimmed mean, the average score would have been 4.89 Trimming the mean reduced the effects of outlier bias in this hypothetical sample by .29 (4.89- 4.6 = .29) MEASURES OF CENTRAL TENDENCY CONTINUED 4 The Median (Mdn). The median is the second quartile or 50th percentile. Statistics Rules: • Arrange the scores into an array. • If numbers of scores (n) are odd, the median is the middle score • If the numbers of scores (n) are even, then average the two middle scores The Mode. It is the value or value category that appears most frequently within a data set. A data set could have no mode or multiple modes. Common terms for the mode: • • • • No mode (Zero mode) Single mode Bimodal (two modes) Multiple modes MEASURES OF VARIATION OR DISPERSION 5 Data spread is called variability or dispersion. Common measures of variation are the range, quartile, mean deviation, variance, and standard deviation. • Range = Maximum – minimum • Quartile is used to locate the 25th, 50th, and 75th percentile of the distribution set • Mean deviation (MD) is a measure of dispersion, which is equal to the mean of the absolute values of the deviation scores. MD = • Variance, also called sum of the squared deviation from the mean, provides an understanding of the spread of scores about 𝑋−𝑋 2 the mean. Variance = 𝑛−1 • Standard deviation (SD) is the most informative measure of variability PROPERTIES OF THE MEAN DEVIATION AND VARIANCE 6 X= 5, 1, 4, 5, 3, 2, AND 5 X = Self-esteem Mean deviation Variance 5 1 4 5 3 2 5 ΣX = 25 = 3.57 5-3.57= +1.43 1-3.57 = -2.57 4-3.57 = +0.43 5-3.57 = +1.43 3-3.57 = -0.57 2-3.57 = -1.57 5-3.57 = +1.43 = 2.04 = 6.60 = .18 = 2.04 = .32 = 2.46 = 2.04 = 15.68/6 = 2.61 Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press COEFFICIENT OF VARIATION (CV) 7 The CV helps researchers to understand the relative variability of a variable when only the sample mean ( ) and its corresponding standard deviation (SD) are present • CV = SD X x 100% Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press