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 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 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 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 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 the mean. Variance = 𝑋− 𝑋 2 𝑛−1 Standard deviation (SD) is the most informative measure of variability

PROPERTIES OF THE MEAN DEVIATION AND VARIANCE X= 5, 1, 4, 5, 3, 2, AND 5 X = Self-esteem Mean deviation Variance 5 1 4 3 2 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 = 2.04 = 6.60 = .18 = .32 = 2.46 = 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) 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