Chapter 3 Measures of Central Tendency

Chapter 3 Measures of Central Tendency I Mode A. Definition: the Score or Qualitative Category that Occurs With the Greatest Frequency 1. Mode (Mo) for the following data, number of required textbooks for Fred’s four classes, is 2. 2 1 2 3

Table 1. Taylor Manifest Anxiety Scores • _______________________________ • (1) (2) f _______________________________ 74 1 73 1 72 0 71 2 70 7 Mo = 69 69 8 68 5 67 2 66 1 65 1 _______________________________ n = 28 _______________________________

II Mean A. Definition: the Mean Is the Sum of Scores Divided by the Number of Scores B. Formula

C. Summation Operator, (Greek capitol sigma) D. Mean Formula for a Frequency Distribution 1. k = number of class intervals

Table 2. Taylor Manifest • Anxiety Scores • _________________ • (1) (2) (3) f _________________ 74 1 74 73 1 73 72 0 0 71 2 142 70 7 490 69 8 552 68 5 340 67 2 134 66 1 66 65 1 65 _________________ n = 28 1,936 _________________

III Median (Mdn) • A. Definition: the Median Divides Data Into Two Groups Having Equal Frequency • 1. If n is odd and the scores are ordered, the median • is the (n + 1)/2th score from either end of the number line. • 2. If n is even, the median is the midway point • between the n/2th score and the n/2 + 1th • score from either end of the number line.

B. Computational Examples 1. Determination of Mdn when n is odd 2. Determination of Mdn when n is even

3. Determination of Mdn when n is even (a) or odd (b), and the frequency of the middle score value is greater than 1

4. Determination of Mdn when n is even and the frequency of the middle score value is greater than 1

C. Computation of Mdn for a Frequency Distribution 1. Formula when scores are cumulated from below Xll = real lower limit of the class interval containing the median i = class interval size n = number of scores fb = number of scores below Xll fi = number of scores in the class interval containing the median

2. Formula for the Mdn when scores are cumulated from above Xul = real upper limit of class interval containing the median fa = number of scores above Xul

Table 3. Taylor Manifest Anxiety Scores _____________________________ __________________________ 74 1 1 73 1 2 72 0 2 71 2 4 70 7 11 69 8 17 19 68 5 9 67 2 4 66 1 2 65 1 1 __________________________ n = 28 __________________________

IV Relative Merits of the Mean, Median, and Mode V Location of the Mean, Median, and Mode in a Distribution

VI Mean of Two or More Means A. Weighted Mean VII Summation Rules A. Sum of a Constant (c)

B. Sum of a Variable (Vi) C. Sum of the Product of a Constant and a Variable D. Distribution of Summation

Chapter 3 Measures of Central Tendency