Frequency Distributions Measures of Central Tendency Finite 9

Frequency Distributions; Measures of Central Tendency Finite 9 -1

• A statistic is a characteristic or measure obtained by using the data values from a sample. • A parameter is a characteristic or measure obtained by using the data values from a specific population. Vocabulary

• The mean is defined to be the sum of the data values divided by the total number of values. • The mean, in most cases, is not an actual data value. Mean

The symbol X represents the sample mean. X is read as " X - bar ". The Greek symbol is read as " sigma" and it means " to sum". X +. . . + X X = n X. = n 1 2 Mean n

The Greek symbol m represents the population mean. The symbol m is read as " mu". N is the size of the finite population. X +. . . + X m= N X. = N 1 2 Mean N

A small company consists of the owner , the manager , the salesperson, and two technicians. The salaries are listed as $50, 000, 20, 000, 12, 000, 9, 000 and 9, 000 respectively. ( Assume this is the population. ) Then the population mean will be X = m N 50, 000 + 20, 000 + 12, 000 + 9, 000 = 5 = $20, 000. Mean

The mean for an ungrouped frequency distributuion is given by (f ·X) X =. n Here f is the frequency for the corresponding value of X , and n = f. Ungrouped

The scores for 25 students on a 4 – point quiz are given in the table. Find the mean score 5 Score , XX Score, 00 FFrrequency, ff 22 11 22 44 12 12 33 44 44 33 5 Frequency Distribution - Example

5 Score , XX Score, 00 FFrrequency, ff 22 ff ? XX 00 11 22 44 12 12 44 24 24 33 44 44 33 12 12 5 f X 52 X= = = 2. 08 n 25 Frequency Distribution - Example

The meanfor a grouped frequency distribution is given by å( f × X ) X =. n Here X is the correspond ing class midpoint. m m Frequency Distribution - Mean

5 CClass 15. 5 --20. 5 FFrrequency, ff 33 20. 5 --25. 5 --30. 5 55 44 30. 5 --35. 5 --40. 5 33 22 5 Frequency Distribution - Example

Table with class midpoints, Xm. 5 CClass FFre quency, ff requency, XXmm 15. 5 --20. 5 --25. 5 33 55 18 18 23 23 25. 5 --30. 5 --35. 5 30. 5 35. 5 44 33 28 28 333 3 35. 5 --40. 5 22 38 38 ff ? XXmm 54 54 115 112 99 99 76 76 5 Frequency Distribution - Example

f X = 54 + 115 + 112 + 99 + 76 = 456 and n = 17. So f X X = n 456 =. . = 2682 17 m m Frequency Distribution - Example

• When a data set is ordered, it is called a data array • The median is defined to be the midpoint of the data array. Vocabulary

• The weights (in pounds) of seven army recruits are 180, 201, 220, 191, 219, 209, and 186. Find the median. • Arrange the data in order and select the middle point. Median - Example

• Data array: 180, 186, 191, 209, 219, 220. • The median, MD = 201. Median - Example

• In the previous example, there was an odd number of values in the data set. In this case it is easy to select the middle number in the data array. • When there is an even number of values in the data set, the median is obtained by taking the average of the two middle numbers Median

• Six customers purchased the following number of magazines: 1, 7, 3, 2, 3, 4. Find the median. • Arrange the data in order and compute the middle point. • Data array: 1, 2, 3, 3, 4, 7. • The median, MD = (3 + 3)/2 = 3. Median - Example

• The ages of 10 college students are: 18, 24, 20, 35, 19, 23, 26, 23, 19, 20. Find the median. • Arrange the data in order and compute the middle point. • Data array: 18, 19, 20, 20 23, 23 23, 24, 26, 35. • The median, MD = (20 + 23)/2 = 21. 5. Median - Example

• For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value. • If n is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above. Median

• LRJ Appliance recorded the number of TVs sold per week over a one-year period. The data is given below. Find the median No. Sets. Sold 11 FFrequenc y requency 44 22 33 99 66 44 55 22 33 Median - Example

• To locate the middle point, divide n by 2; 24/2 = 12. • Locate the point where 12 values would fall below and 12 values will fall above. • Consider the cumulative distribution. • The 12 th and 13 th values fall in class 2. So, MD = 2. 2 Median - Example

No. Sets. Sold FFrequency 11 22 44 99 CCumulative FFrequency 44 13 13 33 44 66 22 19 19 21 21 55 33 24 24 This class contains the 5 th through the 13 th values. Median - Example

The median can be computed from: ( n 2) - cf MD = ( w) + L m f Where n = sum of the frequencies cf = cumulative frequency of the class immediately preceding the median class f = frequency of the median class w = width of the median class L m = lower boundary of the median class Median

5 CClass 15. 5 --20. 5 FFrequency, ff 33 20. 5 --25. 5 --30. 5 55 44 30. 5 --35. 5 --40. 5 33 22 5 Median - Example

5 CClass FFrequency, ff 15. 5 --20. 5 --25. 5 33 55 CCumulative F requency 33 88 25. 5 --30. 5 --35. 5 44 33 12 12 15 15 35. 5 --40. 5 22 17 17 5 Median - Example

• To locate the halfway point, divide n by 2; 17/2 = 8. 5 9. • Find the class that contains the 9 th value. This will be the median class • Consider the cumulative distribution. • The median class will then be 25. 5 – 30. 5 Median - Example

n = 17 cf = 8 f= 4 w = 25. 5 – 20. 5= 5 L m = 255. (n 2) - cf (17/2) – 8 MD = ( w) + L m = (5) + 255. f 4 = 26. 125. Median - Example

• The mode is defined to be the value that occurs most often in a data set. • A data set can have more than one mode. • A data set is said to have no mode if all values occur with equal frequency. Mode

• The following data represent the duration (in days) of U. S. space shuttle voyages for the years 1992 -94. Find the mode. • Data set: 8, 9, 9, 14, 8, 8, 10, 7, 6, 9, 7, 8, 10, 14, 11, 8, 14, 11. • Ordered set: 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 11, 14, 14. Mode = 8. 8 Mode - Example

• Six strains of bacteria were tested to see how long they could remain alive outside their normal environment. The time, in minutes, is given below. Find the mode. • Data set: 2, 3, 5, 7, 8, 10. • There is no mode since each data value occurs equally with a frequency of one. Mode - Example

• Eleven different automobiles were tested at a speed of 15 mph for stopping distances. The distance, in feet, is given below. Find the mode. • Data set: 15, 18, 18, 20, 22, 24, 24, 26. • There are two modes (bimodal) The values are 18 and 24. 24 Mode - Example

• Google Sheets and Histograms Step by Step 1. Get data into a sheet Histograms

1. Get data into a sheet 2. Select data and insert a chart Histograms

1. Get data into a sheet 2. Select data and insert a chart 3. In chart editor, select histogram, insert Histograms

1. 2. 3. 4. Get data into a sheet Select data and insert a chart In chart editor, select histogram, insert Double click title(s) to edit Histograms

1. 2. 3. 4. 5. Get data into a sheet Select data and insert a chart In chart editor, select histogram, insert Double click title(s) to edit To finalize the chart, right click and select Advance Edit - Change Bucket size for more or less bars (8 to 12 is great) - Favorite color can be changed here as well Histograms

Histograms

• Homework questions ask for both histograms and frequency polygons. The latter is only accomplished through reformatting the data and stacking graphs. We will not be doing that for this class. Histograms

• Mean is completed with =AVERAGE(range here) • Median is completed with =MEDIAN(range here) • Mode is completed with =MODE(range here) • If using a frequency table, multiply the frequency by the amount and sum both the frequency column and the value. Divide the sums for the average • Mode does not work for bimodal, so sort ranges to check on mode. Google Sheets

• Pages 415 – 418 • 1, 3, 7 -13 odd, 17 -21 odd, 25 -29 odd, 36, 37, 39, 43 Homework