Interpreting Histograms Chapter 4 Interpreting Histograms Grouped Data are data presented in the form of a histogram, a frequency curve, an interval tally, or a similar display. An interval tally is a list of intervals and their frequency of scores. Calculators can easily find the mean of frequency tables, using the class midpoints and the frequencies. Mean values found from frequency tables will be an approximation of the mean value found using the actual data.

m = class midpoint Mean from a Frequency Table use class midpoint of classes for variable x Estimated value of Calculators can easily find the mean of frequency tables, using the class midpoints and the frequencies. Mean values found from frequency tables will be an approximation of the mean value found using the actual data. m = class midpoint f = frequency N = total number

Standard Deviation from a Frequency Table Estimated value of m = class midpoint f = frequency N = total number m = average Calculators can easily find the mean of frequency tables, using the class midpoints and the frequencies. Mean values found from frequency tables will be an approximation of the mean value found using the actual data.

Find the average mean and standard deviation ACT Scores Find the average mean and standard deviation Interval Frequency [32, 36) 3 [28,32) 6 [24,28) 12 [20,24) 10 [16,20) 8 [12,16) 2 Midpoint 34 30 26 22 18 14 Product 102 180 312 220 144 28 sum = 41 sum = 986 average mean is 986/41 = 24.04

Interval Frequency [32, 36) 3 [28,32) 6 [24,28) 12 [20,24) 10 [16,20) 8 [12,16) 2 (mi - ) 9.96 5.96 1.96 -2.04 -6.04 -10.04 Midpoint 34 30 26 22 18 14 (mi - )2 99.20 35.52 3.84 4.16 36.48 100.8 fi(mi - )2 297.6 213.13 46.099 41.616 291.85 201.16 sum = 1091.9

Weighted Mean  (w • x) x =  w A weighted mean of a group of scores is a mean computed in such a way that the frequency, or relative importance, of each score is taken into consideration x = w  (w • x)  Used when the data values are assigned different weights, such as grades received and the computation of a GPA. Used when the data values are assigned different weights, such as grades received and the computation of a GPA.

Find the Grade Point Average Class Credit Grade Math 4 B+ History 3 B Physics C English 5 B- (4)(3.3)+3(3.0)+4(2.0)+5(2.6) = 43.2 43.2/16 = 2.7

(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) page 79 of text x

(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) 68% within 1 standard deviation Some student have difficulty understand the idea of ‘within one standard deviation of the mean’. Emphasize that this means the interval from one standard deviation below the mean to one standard deviation above the mean. 34% 34% x - s x x + s

(applies to bell-shaped distributions) The Empirical Rule (applies to bell-shaped distributions) 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 13.5% 13.5% x - 2s x - s x x + s x + 2s

The Empirical Rule x - 3s x - 2s x - s x x + s x + 2s x + 3s (applies to bell-shaped distributions) 99.7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation These percentages will be verified by the concepts learned in Chapter 5. Emphasize the Empirical Rule is appropriate for data that is in a BELL-SHAPED distribution. 34% 34% 2.4% 2.4% 0.1% 0.1% 13.5% 13.5% x - 3s x - 2s x - s x x + s x + 2s x + 3s

99.7% of data are within 3 standard deviations of the mean IQ Scores have an average of 100 with a standard deviation of 10 70 80 90 100 120 130 110 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean 0.1% 2.4% 13.5% These percentages will be verified by the concepts learned in Chapter 5. Emphasize the Empirical Rule is appropriate for data that is in a BELL-SHAPED distribution.

Chebyshev’s Theorem applies to distributions of any shape. the proportion (or fraction) of any set of data lying within K standard deviations of the mean  is always at least 1 - 1/K2 , where K is any  positive number greater than 1. at least 3/4 (75%) of all values lie within 2  standard deviations of the mean. at least 8/9 (89%) of all values lie within 3  standard deviations of the mean. Emphasize Chebyshev’s Theorem applies to data that is in a distribution of any shape - that is, it is less prescriptive than the Empirical Rule. page 80 of text

Find the average mean and standard deviation ACT Scores Find the average mean and standard deviation Interval Frequency [32, 36) 3 [28,32) 6 [24,28) 12 [20,24) 10 [16,20) 8 [12,16) 2 Midpoint 34 30 26 22 18 14 Product 102 180 312 220 144 28 sum = 41 sum = 986 average mean is 986/41 = 24.04  = 5.16

What interval, centered around the mean in which approximately 68% of the ACT scores? 95% of the ACT scores? 99% of the ACT scores? [18.9,29.2] a. 24.04  5.16 b. 24.04  2(5.16) [13.72,34.36] c. 24.04  3(5.16) [8.58,39.44] [8.58,36] Round to 36

Using Chebyshev’s Theorem, what interval contains 3/4 of the ACT scores? 8/9 of the ACT scores? 15/16 of the ACT scores? a. 24.04  2(5.16) [13.74,34.12] b. 24.04  3(5.16) [8.58,36] 24.04  4(5.16) [3.12,36] c.

Estimating the mode L lower limit of the modal interval b L w L lower limit of the modal interval w is the modal interval width a and b are the differences in frequencies

Estimating the median L lower limit of the median interval w is the modal interval width N is the total number of scores F is the sum of the frequencies up to but not including the median interval f is the frequency of the median interval

Find the estimated mode and median 2 4 6 8 10 5 9 7 8 4 = 4.333 = 5.714

Find the estimated mode and median 1 3 5 7 9 6 11 8 9 5 = 3.25 = 4.625

A control chart is a graph that can be used to Control Charts A control chart is a graph that can be used to indicate how a series of new scores compare with a historical based mean and standard deviation mean + 1 standard deviation mean mean - 1 standard deviation