Statistics

Descriptive Statistics Inferential Statistics

Descriptive statistics Used to describe a set of data Inferential statistics Used to draw inferences about a population based on data obtained from a sample. Parameter Statistic Descriptive Statistics Inferential Statistics

Scales of Measurement Nominal scales (Categorical data) Ordinal scales Interval scales Ratio scales Descriptive statistics  Used to describe a set of data Inferential statistics Used to draw inferences about a population based on data obtained from a sample. population - by definition any entire group of events or people  Parameter - Greek letters used . Used fto describe populations Statistic - used to describe samples

Distributions IQ Scores 113, 117, 84, 103, 105, 108, 115, 126, 103, 112, 106, 111, 113, 90, 107, 76, 109, 97, 103, 97, 89, 101, 93, 99, 115, 94, 94, 94, 108, 97, 95, 87, 107, 97, 88, 99, 97, 82, 112, 91, 107, 93, 83, 99, 100, 92, 102, 93, 100, 83, 125, 96, 84, 113, 113, 96, 97, 88, 100, 95, 86, 77, 80, 125, 110, 114, 121, 102, 90, 103, 91, 103, 109, 104, 79, 100, 85, 76, 115, 105, 95, 88, 93, 111, 107, 108, 98, 84, 123, 107, 118, 93, 101, 92, 99, 92, 136, 121, 97, 101, 102, 114, 103, 81, 71, 104, 85, 106, 120, 108, 112, 131, 94, 104, 121, 123, 91, 103, 93, 102 N = 120 Scales of Measurement Nominal scales (Categorical data) not really scales Ordinal scales cannot really talk about differences between scale points Interval scales equal intervals. Ratio scales  type of scales dtermines type of analysis

IQ Scores - Ranked 71, 76, 76, 77, 79, 80, 81, 82, 83, 83, 84, 84, 84, 85, 85, 86, 87, 88, 88, 88, 89, 90, 90, 91, 91, 91, 92, 92, 92, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 95, 95, 95, 96, 96, 97, 97, 97, 97, 97, 97, 97, 98, 99, 99, 99, 99, 100, 100, 100, 100, 101, 101, 101, 102, 102, 102, 102, 103, 103, 103, 103, 103, 103, 103, 104, 104, 104, 105, 105, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 109, 109, 110, 111, 111, 112, 112, 112, 113, 113, 113, 113, 114, 114, 115, 115, 115, 117, 118, 120, 121, 121, 121, 123, 123, 125, 125, 126, 131, 136 N = 120 Distributions IQ Scores 113, 117, 84, 103, 105, 108, 115, 126, 103, 112, 106, 111, 113, 90, 107, 76, 109, 97, 103, 97, 89, 101, 93, 99, 115, 94, 94, 94, 108, 97, 95, 87, 107, 97, 88, 99, 97, 82, 112, 91, 107, 93, 83, 99, 100, 92, 102, 93, 100, 83, 125, 96, 84, 113, 113, 96, 97, 88, 100, 95, 86, 77, 80, 125, 110, 114, 121, 102, 90, 103, 91, 103, 109, 104, 79, 100, 85, 76, 115, 105, 95, 88, 93, 111, 107, 108, 98, 84, 123, 107, 118, 93, 101, 92, 99, 92, 136, 121, 97, 101, 102, 114, 103, 81,  71, 104, 85, 106, 120, 108, 112, 131, 94, 104, 121, 123, 91, 103, 93, 102 N = 120

Frequency distribution of IQ scores Cumulative Value Frequency Percent Percent 71 1 .8 .8 76 2 1.7 2.5 77 1 .8 3.3 79 1 .8 4.2 80 1 .8 5.0 81 1 .8 5.8 82 1 .8 6.7 83 2 1.7 8.3 84 3 2.5 10.8 85 2 1.7 12.5 86 1 .8 13.3 IQ Scores - Ranked 71, 76, 76, 77, 79, 80, 81, 82, 83, 83, 84, 84, 84, 85, 85, 86, 87, 88, 88, 88, 89, 90, 90, 91, 91, 91, 92, 92, 92, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 95, 95, 95, 96, 96, 97, 97, 97, 97, 97, 97, 97, 98, 99, 99, 99, 99, 100, 100, 100, 100, 101, 101, 101, 102, 102, 102, 102, 103, 103, 103, 103, 103, 103, 103, 104, 104, 104, 105, 105, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 109, 109, 110, 111, 111, 112, 112, 112, 113, 113, 113, 113, 114, 114, 115, 115, 115, 117, 118, 120, 121, 121, 121, 123, 123, 125, 125, 126, 131, 136 N = 120 87 1 .8 14.2 88 3 2.5 16.7 89 1 .8 17.5 90 2 1.7 19.2 91 3 2.5 21.7 . . . . 136 1 .8 100.0 Total 120 100

Grouped Frequency Distribution of IQ scores Mid- Cumulative Value F Percent point Percent 70 - 74 72 1 .8 .8 75 -79 77 4 3 .3 4 .2 80 - 84 82 8 6 .6 10 .8 85 - 89 87 8 6 .6 17 .5 90 - 95 92 18 15 .0 32 .5 95 - 99 97 17 14 .2 46 .7 100 -104 102 21 17 .5 64 .2 105 - 109 107 15 12 .5 76 .7 IQ Scores - Ranked 71, 76, 76, 77, 79, 80, 81, 82, 83, 83, 84, 84, 84, 85, 85, 86, 87, 88, 88, 88, 89, 90, 90, 91, 91, 91, 92, 92, 92, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 95, 95, 95, 96, 96, 97, 97, 97, 97, 97, 97, 97, 98, 99, 99, 99, 99, 100, 100, 100, 100, 101, 101, 101, 102, 102, 102, 102, 103, 103, 103, 103, 103, 103, 103, 104, 104, 104, 105, 105, 106, 106, 107, 107, 107, 107, 107, 108, 108, 108, 108, 109, 109, 110, 111, 111, 112, 112, 112, 113, 113, 113, 113, 114, 114, 115, 115, 115, 117, 118, 120, 121, 121, 121, 123, 123, 125, 125, 126, 131, 136 N = 120 110 - 114 112 12 10 .0 86 .7 115 - 119 117 5 4 .2 90 .8 120 - 124 122 6 5 .0 95 .8 125 - 129 127 3 2 .5 98 .3 130 - 134 132 1 .8 99 .2 135 - 139 137 1 .8 100 .0 N 120

Grouped Frequency Distribution of IQ scores Percent is term used in SPSS - also known as Relative Frequency Cumulative percent = Cumulative frequency distribution

Grouped Frequency Distribution of IQ scores Percent is term used in SPSS - also known as Relative Frequency Cumulative percent = Cumulative frequency distribution

Grouped Frequency Distribution of IQ scores Percent is term used in SPSS - also known as Relative Frequency Cumulative percent = Cumulative frequency distribution

Grouped Frequency Distribution of IQ scores Percent is term used in SPSS - also known as Relative Frequency Cumulative percent = Cumulative frequency distribution

Measures of Central Tendency Mode The most common score Shapes of Distributions Kurtosis

Grouped Frequency Distribution of IQ scores Mid- Cumulative Value F Percent point Percent 70 - 74 72 1 .8 .8 75 -79 77 4 3 .3 4 .2 80 - 84 82 8 6 .6 10 .8 85 - 89 87 8 6 .6 17 .5 90 - 95 92 18 15 .0 32 .5 95 - 99 97 17 14 .2 46 .7 100 -104 102 21 17 .5 64 .2 105 - 109 107 15 12 .5 76 .7 Measures of Central Tendency Mode The most common score Is the only measure that can be used with Nominal data. 110 - 114 112 12 10 .0 86 .7 115 - 119 117 5 4 .2 90 .8 120 - 124 122 6 5 .0 95 .8 125 - 129 127 3 2 .5 98 .3 130 - 134 132 1 .8 99 .2 135 - 139 137 1 .8 100 .0 N 120

Measures of Central Tendency Median (Mdn) The score that corresponds to the point at or below which 50% of the scores fall. (The 50th percentile.) Mode = 102

2 N + 1 Median location = Sample A:26811141620 Mdn = 11.0 Sample B:2681114162051 Mdn = 12.5 Measures of Central Tendency Median (Mdn) The score that corresponds to the point at or below which 50% of the scores fall. (The 50th percentile.) 2 N + 1 Median location =

Grouped Frequency Distribution of IQ scores Mid- Cumulative Value F Percent point Percent 70 - 74 72 1 .8 .8 75 -79 77 4 3 .3 4 .2 80 - 84 82 8 6 .6 10 .8 85 - 89 87 8 6 .6 17 .5 90 - 95 92 18 15 .0 32 .5 95 - 99 97 17 14 .2 46 .7 100 -104 102 21 17 .5 64 .2 105 - 109 107 15 12 .5 76 .7 Mdn = 100.50 110 - 114 112 12 10 .0 86 .7 115 - 119 117 5 4 .2 90 .8 120 - 124 122 6 5 .0 95 .8 125 - 129 127 3 2 .5 98 .3 130 - 134 132 1 .8 99 .2 135 - 139 137 1 .8 100 .0 N 120

å Mean ( 0, : or M) 77 X = = 11 . 7 128 X = = 16 8 X X = N The average score å X ( 3 = capital Sigma = add up all the numbers) X = N Sample A: 2 6 8 11 14 16 20 77 X = = 11 . (Mdn = 11.0 ) A 7 Sample B 2 6 8 11 14 16 20 51 128 ( Mdn = 12.5 ) X = = 16 B 8 Outlier

' ' Properties of the mean X X - X - 2 2 - 11 -9 6 6 - 11 -4 8 8 - 11 X X - X - 2 2 - 11 -9 6 6 - 11 -4 8 8 - 11 -3 11 11 - 11 Mean ( í,  or M) The average score X bar used by Howell and many others. THis is for a sample . Use mu for population. M is APA format. ERROR - Mdn for second example is 12.5 14 14 - 11 3 16 16 - 11 4 20 20 - 11 9 9 ' ' X - X - = 0 = 0

Scores Sample A 10 12 15 18 20 Sample B 2 8 15 22 28 Sample C 15 15 15 Properties of the mean Mean vs median Median not affected by extreme values - only interested in the position of the scores. However it cannot be used in further calculations. Mean is most common and most useful but can be distorted  Mean and median in relation to skewness.

Scores Mdn M Sample A 10 12 15 18 20 15 15 Sample B 2 8 15 22 28 15 15 Sample C 15 15 15 15 15 15 15 Measures of Variability

Measures of Variability Range The scale distance between the largest and the smallest scores. Scores Mdn M Sample A 10 12 15 18 20 15 15 Sample B 2 8 15 22 28 15 15 Sample C 15 15 15 15 15 15 15 Measures of Variability RangeA = 20 - 10 = 10 RangeB = 28 - 2 = 26 RangeC = 15 - 15 = 0

Interquartile Range Is the distance between the score occuring at the 25th percentile and the score occuring at the 75th percentile. Range The scale distance between the largest and the smallest scores.

å ' ' Mean (Average) Deviation ( X - X ) N X X - X - 2 2 - 11 -9 6 X X - X - 2 2 - 11 -9 6 6 - 11 -4 8 8 - 11 -3 11 11 - 11 Interquartile Range Is the distance between the score occuring at the 25th percentile and the score occuring at the 75th percentile. IQ data 108.75 - 93.00 = 15.75 14 14 - 11 3 16 16 - 11 4 20 20 - 11 9 9 ' ' X - X - = 0 = 0

å Mean Absolute Deviation X - X N ' ' X X - * X - * 2 2 - 11 9 6 X X - * X - * 2 2 - 11 9 6 6 - 11 5 Mean (Average) Deviation 8 8 - 11 3 11 11 - 11 14 14 - 11 3 16 16 - 11 4 20 20 - 11 9 9 ' ' X - X - = 32 = 32

å å Variance (for population) ( X - X ) s = N ( X - X ) 68 s = = = 13 2 s 2 = N Sample X X - (X - ) 2 A 10 -5 25 12 -3 9 15 18 3 9 20 5 25 Mean Absolute Deviation G = 65 68 = 15 å ( X - X ) 2 68 s 2 = = = 13 . 60 N 5

å å Standard Deviation (for population) ( X - X ) 68 s = = = 13 . 6 = 2 s = N Sample X X - (X - ) 2 A 10 -5 25 12 -3 9 15 18 3 9 Variance (for population) Stricktly speaking you should not use this formula with a sample as in this example but it is to show you how to do the calculation. 20 5 25 G = 65 68 = 15 å ( X - X ) 2 68 s = = = 13 . 6 = 3 . 69 N 5

å å Variance (for sample) ( X - X ) s = N - 1 ( X - X ) 68 s = = = 17 2 s 2 = N - 1 Sample X X - (X - ) 2 A 10 -5 25 12 -3 9 15 18 3 9 20 5 25 Standard Deviation (for population) G = 65 68 = 15 å ( X - X ) 2 68 s 2 = = = 17 . 00 N - 1 4

å å Standard Deviation (for sample) Sample X X - (X - ) A 10 -5 25 12 = N - 1 Sample X X - (X - ) 2 A 10 -5 25 12 -3 9 15 18 3 9 Variance (for sample)  Samples as estimators of populations - slightly more conservative approach.  20 5 25 G = 65 68 = 15 å ( X - X ) 2 68 s = = = 17 . = 4 . 12 N - 1 4

Degrees of freedom Standard Deviation (for sample)

Boxplot of IQ Scores Degrees of freedom