Measures of Dispersion How much scores are spread out Let’s assume that you are thinking about the age of two groups of subjects. • Group A: 34.5 years • Group B - 34.5 years Range: • The distance between the lowest and highest score. • Only identifies the minimum and maximum values. Range of each group • Group A: 22 – 50 • Group B: 22 - 50 Standard Deviation The average amount that scores vary from the mean. As the standard deviation gets larger there is a greater amount of variation between scores. Standard deviation of the groups (Remember the mean = 34.5 years) • Group A: standard deviation - 5 years – In general the subjects’ ages fall between 29.5 years and 39.5 years • Group B: standard deviation – 10 years – In general the subjects’ ages fall between 24.5 years and 45.5 years. Group A Group B S : s y m b o l fo r s ta n d a rd d e v ia tio n o f a s a m p le . : s y m b o l fo r s ta n d a rd d e v ia tio n o f th e p o p u la tio n . For a sample: x x 2 s n Average of individual scores X X n A v e ra g e o f d iffe re n c e s b e tw e e n s c o re s a n d th e m ean. X X ( X – X) N n H o w e v e r: ( X – X) = 0 So… B y s q u a rin g th e d iffe re n c e s , th is p ro b le m is a v o id e d . S 2 ( X – XX ) X N 2 2 n T h is is k n o w n a s th e v a ria n c e . T h e s y m b o ls fo r v a ria n c e a re : S 2 fo r th e s a m p le 2 fo r th e p o p u la tio n . X X 2 S 2 n S S 2 X 2 N For a sample: x x 2 s n X 30 28 25 23 22 18 X X = ------N s N= 6 x x 2 n X 30 28 25 23 22 18 146 X X n X = 146 X 146 n6 146 X 6 X 24.66 For a sample: x x 2 s n X 30 28 25 23 22 18 X–X 5.67 3.67 .67 -1.33 -2.33 -6.33 (30 – 24 .33 ) For a sample: x x 2 s n X 30 28 25 23 22 18 X–X 5.67 3.67 .67 -1.33 -2.33 -6.33 (X – X) 32.15 13.47 .45 1.76 5.43 40.07 93.33 2 X X 2 93.33 X X 2 S 2 n 93.33 6 S 2 S 15.55 S 2 S 3.94 C o m p u ta tio n a l F o rm u la S =S ( X )2 X 2 2 ( XX ) - N n ------------------n N 2 X 30 28 25 23 22 18 N=6 X = 146 (X)2 = 1462 (X)2 = 21316 C o m p u ta tio n a l F o rm u la S =S ( X )2 X 2 2 ( XX ) - N n ------------------n N 2 X 30 28 25 23 22 18 X2 900 784 625 529 484 324 3646 = X2 C o m p u ta tio n a l F o rm u la S =S ( X )2 X 2 2 ( XX ) - N n ------------------n N 2 21316 3646 2 6 S 6 3646 3552.67 S 6 2 93.33 S 6 2 S 15.55 2