1. Measures Of Dispersion : Grouped Data VARIANCE AND STANDARD DEVIATION

2. Variance  Formulas for population variance and standard deviation of grouped data

3.  Formulas for sample variance and standard deviation of grouped data

4. Example 1 The table below show the Frequency Distribution of 60 Years of Unemployment Data for Canada (Grouped Data) Calculate population Variance and Standard Deviation for this data

5. Solution  1 st Method Using Original Formula

6.  2 nd Method using computational formula

7. Example 2 Compute the variance, and standard deviation on the following sample data.

8. Solution

9. Exercise The radio music listener market is diverse. Listener formats might include adult contemporary, album rock, top 40, oldies, rap, country and western, classical, and jazz. In targeting audiences, market researchers need to be concerned about the ages of the listeners attracted to particular formats. Suppose a market researcher surveyed a sample of 170 listeners of country music radio stations and obtained the following age distribution. What are the variance and standard deviation of the ages of country music listeners? Answer: 58.483, 7.647

10. Quartile  Divide data sets into fourths or four equal parts.

11. Example  The positions are integers Example: 5, 8, 4, 4, 6, 3, 8 (n=7)  1. Put them in order: 3, 4, 4, 5, 6, 8, 8  2. Calculate the quartiles 3, 4, 4, 5, 6, 8, 8 Q1=4Q2=5Q3=8

12.  The positions are not integers Example: 5, 12, 10, 4, 6, 3, 8, 14 (n=8)  1. Put them in order: 3, 4, 5, 6, 8, 10, 12, 14  2. Calculate the quartiles 3, 4, 5, 6, 8, 10, 12, 14 2.25 th place 4.5 th place 6.75 th place Q1=4.25 Q2=7 Q3=11.5

13. Exercise The following data represent the number of inches of rain in Chicago during the month of April for 10 randomly years. 2.473.973.944.11 5.22 1.144.023.411.850.97 Determine the quartiles. Answer: Q 1 =1.495Q 2 =3.675 Q 3 =4.0425

14. Box-and-Whisker Plots (Boxplots)  Another way to describe a distribution of data is by using a box and whisker plot.  A box-and-whisker plot, sometimes called a box plot, is a diagram that utilizes the upper and lower quartiles along with the median and the two most extreme values to depict a distribution graphically.

15.  The box-and-whisker plot is determined from five specific numbers sometimes referred to as the five-number summary. five-number summary 1. The median (Q 2 ) 2. The lower quartile (Q 1 ) 3. The upper quartile (Q 3 ) 4. The smallest value in the distribution 5. The largest value in the distribution

16. Steps to constructs Boxplots 1. Find median (Q 2 ), the lower quartile (Q 1 ) and the upper quartile (Q 3 ) 2. A box is drawn around the median with the lower and upper quartiles as the box endpoints. 3. Compute the interquartile range (IQR) is computed by using Q 3- Q 1 4. Determine the fences. Fences serve as cutoff points for determining outliers. If data value is less than the lower fence or greater than the upper fence, considered outlier. 5. Draw a line from Q 1 to the minimum data value that is larger than the lower fence. Draw a line from Q 3 to the maximum data value that is smaller than the upper fence.

19. Exercise