5.6 MEAN, MEDIAN, MODE Textbook Reference Section 12.1 CLAST

SECTION 5.6 Mean, Median, Mode 221

5.6 MEAN, MEDIAN, MODE Textbook Reference Section 12.1 CLAST OBJECTIVES " Determine the mean, median, and mode " Recognize properties and interrelations among the mean, median, and mode " Infer relations and make accurate predictions from studying statistical data " Interpret real world data involving frequency and cumulative frequency tables

Mean, median, and mode are called measures of central tendency because each describes the center of the data sample.

Mean (Average) The mean or average is a calculation and is found by dividing the sum of the data values by the number of data values. Sum of data values Mean = Number of data values Median The median is the middle position when data values are ranked from low to high. To find the median rank the data in ascending order. If there is an odd number of data values, then the median is the value that lies in the middle position. If there is an even number of data value, then the median is the average of the two middle values.

Mode The mode is the most frequently occurring (popular) value.

Example Solution a) Find the mean, median, and mode There are 10 elements in the set.

of the following set: Mean: {5, 20, 35, 20, 50, 45, 20, 10, 25, 45} 5 + 20 + 35 + 20 + 50 + 45 + 20 + 10 + 25 + 45

275 = = 27.5 10

Median: 5, 10, 20, 20, 20, 25, 35, 45, 45, 50 Since we have 10 data values, the middle positions are the 5 th and 6 th positions. Note there are four data values below the 5 th position and 4 data values above 6 th positions. 20 and 25 are in the 5 th and 6 th positions. When we average 20 and 20 + 25 45 25, we get = = 22.5 5 2 Thus, the median is 22.5.

Mode: The mode is 20 as it appears most frequently.

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Check Your Progress 5.6 1. Consider the scores of nine students: {84, 88, 52, 87, 70, 66, 95, 84, 71}. Find the mean, median, and mode.

2. Consider the following set of babies’ birth-weight (in pounds): {6.5, 7, 10, 8, 7, 7.5, 8, 7}. Find the mean, median, and mode.

Comparing Mean, Median, and Mode Mean Median Consider three cases. Mode

Case 1: Symmetric

123456789

Median Mode Mean Case 2: Right - Skewed

123456

Mode Case 3: Left - Skewed

Median

Mean

123456

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Examples Solutions b) The graph below shows the number of hours that college students spend in the game room per week with the years spent in college. Compare the mean, median, and mode. The distribution is symmetric. Thus the mean, median, and mode are equal. s Hour

12345 Years in College

c) The graph below shows the distribution of scores on an algebra quiz. Compare the mean, median, and mode. The distribution is right-skewed. This 25 indicates that the mode is less than the 20 median, which is less than the mean. 15 10

Frequency 5 0 65 70 75 80 85 90 95 Algebra Quiz Scores

d) Half of the students scored 90 on an English test. Most of the remaining students scored 70

and the last few scored 30. Compare the mean, median, and mode. The distribution is skewed left. This indicates that the mean is less than the median, which is less than the mode.

30 70 90

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Check Your Progress 5.6 3. The ounces of milk consumed per day by 6-month old babies is shown in the graph below. Compare the mean, median, and mode.

6 5 4 3 2 1

Number of Babies 0 22 24 26 28 30' 32 34 Ounces of Milk

4. The distribution of placement exam scores are shown below. Discuss the mean, median, and mode.

0 95 90 85 80 75 70 65 60 Placement Test Scores

5. More than half of the shoes in a store are priced at $34.00. Most of the others are priced at $45.00, and a few are priced at $61.00. Discuss the mean, median, and mode.

6. Half of the scores on a History test were 95. Most of the remaining scores were 75. A few were 40. Discuss the mean, median, and mode.

Applications of Mean, Median, and Mode

Mode: The category with the highest frequency, percent, or proportion. Mean: Multiply each category by its proportion and add. Median: Organize categories in ascending order in one column. Write the corresponding frequencies or proportions in an adjacent column. Starting at the top, add the frequencies or proportions up to the fifty-percent point. The corresponding category will be the median.

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Examples Solutions e) The table below shows the proportion of Mean: students taking a number of college-level = 0(0.02) + 1(0.02) + 2(0.11) classes this semester. Find the mean, + 3(0.23) + 4(0.54) + 5(0.08) median, and mode. = 0 + 0.02 + 0.22 + 0.69 + 2.16 + 0.4 No. of classes Proportion = 3.49 0 0.02 1 0.02 Median: (Find the 50 % mark) 2 0.11 0.02 + 0.02 + 0.11 + 0.23 = 0.38 Å 38 % 3 0.23 0.38 + 0.54 = 0.92 Å 92 % 4 0.54 This indicates that the 50 % - mark falls in 5 0.08 the “ 4 Classes ” category. Thus the median is 4 classes.

Mode: The mode is 4 classes because the “ 4 Classes” category has the largest proportion. f) In the table below, the scores on a Spanish test are listed with the corresponding percentile rank. What percent of students scored between 60 and 80? What percent scored above 80? What percent scored below 60?

Score Percentile Rank Between 60 and 80: 93 % – 34 % = 59 % 90 99 80 93 Above 80: 99 % – 93 % = 6 % 70 72 60 34 Below 60: 34 % 40 16 50 5

Check Your Progress 5.6 For Questions 7 and 8, use the information found in the table below. The table shows the results of a survey in which people were asked the number of times per week that they shop at a grocery store. No. of Trips to Grocery Store Percent Men Percent Women 0 18 7 1 45 12 2 12 53 3 17 13 4 8 15 7. Find the median number of trips to the grocery store for men.

8. Find the mean number of trips to the grocery store for women.

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For Questions 9 and 10, use the information in the table below, which shows the distribution of the total family income for a certain county for the year 2000.

Income Level ($) Percent of Families 0 – 9,999 12 10,000 – 19,999 14 20,000 – 29,999 16 30,000 – 39,999 13 40,000 – 49,999 13 50,000 – 59,999 9 60,000 – 69,999 9 70,000 and over 14

9. What percentage of families has incomes of at least $30,000?

10. What percentage of families has incomes between $20,000 and $49,999?

See If You Remember SECTIONS 5.1 – 5.5

1. How many 5-digit numbers can be formed from the digits 0 – 9, if digits cannot be repeated?

2. Seven candidates are running for three city council seats. How many different groups can be formed?

3. What are the odds in favor of randomly drawing a green ball from a box that contains 2 yellow, 3 red, 5 green, and 6 white balls?

4. A fair die is rolled three times. What is the probability of getting a three, then a two, and then a one?