Statistics—Chapter 6 Measures of Dispersion Reading Assignment: p , , 210, , ,

Measures of dispersion 3 Measures of central tendency (Mean, median mode ) are values around which the respondents “gather” Measures of dispersion show how tightly packed (or spread out) the respondents are in relation to one or more of the measures of central tendency Article, p.196

Variability 5 Variability—the degree of variation in the responses to a variable The more homogeneous a distribution of responses, the less variability (or variation) there is in the responses to a variable The more heterogeneous a distribution of responses, the more variability (or variation) there is in the responses to a variable

Measures of relative standing 5 Median Equal subdivisions of data Values which divide into 4 equal parts: quartiles Q1, Q2, Q3 (Q2=median) 10 equal parts: deciles D1, D2,…,D9 (D5=median) 100 equal parts: percentiles P1, P2, …, P99 (P50=median)

Measures of relative standing—pth percentile 2 For any set of n measurements (arranged in ascending or descending order), the pth percentile is a number such that p% of the measurements fall below the pth percentile and (100-p)% fall above it or equal it Ex/120 test scores: ; ; ; ; ; ; th percentile: 75N/100= 75(120)/100=90 Hence, the first 90 scores are in the 75 th percentile; there are 79 scores between 30 and 79, so need 11 of the 32 from the class (10)/32= =83.34 So 75% of scores are below 83 and 25% are above

Range and interquartile range 3 Range= Highest – lowest; Aaron HR: range=47-10=37 Can do in SPSS Range in this case: his home run totals ranged from a low of 10 to a high of 47 Interquartile range (IQR) is a measure of dispersion that tells about the distribution of responses to a variable in relation to the median It is used with variables that are ordinal or numerical and gives the range of the values comprising the middle 50% of the data set

IQR 4 Dividing the set into 4 parts, with each part having the same number of cases See p. 204—Students/earners IQR= Q3-Q1, where Q3 is the third quartile value, and Q1 is the value at the first quartile [P. 203, table 6.3] Q1: N=2841; N+1=2842; (N+1)(0.25)=(2842)(0.25)=710.5, so the value of the first quartile is associated with response 1); similarly, Q3: response 2)

IQR (cont) 5 P. 207, Skills 2 Do for Aaron HR data (aaronIQR) Do in SPSS: A/DS/F/S/Q Q1=26, Q3=44; IQR=44- 26=18;check cumulative % Picture—Boxplot—G/BP

Boxplot terms 5 Outliers: unusually high or low values for a variable Extreme values: exceptionally high or low values in relation to the interquartile range Outlier in the bottom range: value that is <= Q1-(IQR*1.5) Outlier in the upper range: value that is >= Q3+(IQR*1.5) P. 215—follow book example for educ

Mean Deviation 3 Mean deviation: average of the deviations from the mean Subtract mean from each score in the data set Find abs val of each dev. From the mean Add up results Divide the sum of the deviations from the mean by N to get average of the deviations from the mean Ex/ p. 217 Excel:aaron HR

4 Variance Like the mean deviation, but we square each difference instead of taking the absolute value More useful than mean deviation in that it leads to the standard deviation, which is its square root (demonstrate) Ex/ p.218 Aaron excel

4 Standard deviation Uses the mean as a point of comparison Explanation, p. 219 Formula, p. 220 Skills 5,6 (hw: 7, 8) Chart, p. 233

4 Chebyshev’s Rule: any sample, regardless of the f.d.shape It is possible that very few of the measurements fall within 1 std. Dev. Of the mean At least ¾ of the measurements will fall within 2 std dev of the mean (m-2s, m+2s) At least 8/9 of the measurements will fall within 3 std dev of the mean (m-3s, m+3s) Generally, at least 1/k^2 of the measurements will fall within k std dev of the mean (m-ks, m+ks), k>1 Explanation, p. 219 Formula, p. 220 Skills 5-8 Chart, p. 233

3 Empirical Rule –frequency distributions are mound shaped Apprx. 68% of the measurements will fall within 1 std dev of the mean (m-s, m+s) Approx. 95% of the measurements will fall within 2 std dev of the mean (m-2s, m+2s) Essentially all the measurements will fall within 3 std dev of the mean (m-3s, m+3s)

homework Skills 5-8 Hand in 6/17 p. 245 SPSS 1,2