Holt McDougal Algebra 1 10-2 Frequency and Histograms A measure of central tendency describes the...

Holt McDougal Algebra 1

10-2 Frequency and Histograms

A measure of central tendency describes the center of a set of data. Measures of central tendency include the mean, median, and mode.

• The mean is the average of the data values, or the sum of the values in the set divided by the number of values in the set.

• The median the middle value when the values are in numerical order, or the mean of the two middle numbers if there are an even number of values.



The range of a set of data is the difference between the least and greatest values in the set. The range describes the spread of the data.

• The mode is the value or values that occur most often. A data set may have one mode or more than one mode. If no value occurs more often than another, we say the data set has no mode.



Additional Example 1: Finding Mean, Median, Mode, and Range of a Data Set

mean:

Write the data in numerical order.

Add all the values and divide by the number of values.

There are an even number of values. Find the mean of the two middle values.

median: 150, 150, 156, 156, 161, 163The median is 156.

The weights in pounds of six members of a basketball team are 161, 156, 150, 156, 150, and 163. Find the mean, median, mode, and range of the data set.



Additional Example 1 Continued

150, 150, 156, 156, 161, 163

modes: 150 and 156 150 and 156 both occur more often than any other value.range: 163 – 150 = 13



Check It Out! Example 1 The weights in pounds of five cats are 12, 14, 12, 16, and 16. Find the mean, median, mode, and range of the data set.

12, 12, 14, 16, 16

Write the data in numerical order.

Add all the values and divide by the number of values.

median: 12, 12, 14, 16, 16The median is 14.

There are an odd number of values. Find the middle value.



Check It Out! Example 1 Continued

mode: 12 and 16

range: 12 – 16 = 4

The data set is bi-modal as 12 and 14 both occur twice.

The weights in pounds of five cats are 12, 14, 12, 16, and 16.Find the mean, median, mode, and range of the data set.



A stem-and-leaf plot arranges data by dividing each data value into two parts. This allows you to see each data value.

The last digit of a value is called a leaf.

The digits other than the last digit of each value are called a stem.

Key: 2|3 means 23

The key tells you how to read each value.



Example 2A: Reading a Stem-and-Leaf Plot

The stem-and-leaf plot below shows the numbers of defective widgets in batches of 1000 are given below. Use the stem-and-leaf plot to answer the following questions.

Number of DefectiveWidgets per Batch

Stem Leaves

0 8 8 9 91 2 3 3 4 5 92 0 1





Stem Leaves

0 8 8 9 91 2 3 3 4 5 92 0 1

A. What is the median number of defective widgets per batch?

B. What is the mean number of defective widgets per batch?





Stem Leaves

0 8 8 9 91 2 3 3 4 5 92 0 1

C. How many orders had less than 15 defective widgets?

D. About what percent of the batches had fewer than 10 defective widgets?



Example 2B Reading a Stem-and-Leaf Plot

Football StateChampionship Scores

Team A Team B3 7

9 5 2 2 4 8 8 9 8 6 5 5 2 7 8

5 1 6 0 1 3 2 7

The season’s scores for the football teams going to the state championship are given in the stem-and-leaf plot below. Use the data to answer the following questions



Example 2B Reading a Stem-and-Leaf Plot

Football StateChampionship Scores

Team A Team B3 7

9 5 2 2 4 8 8 9 8 6 5 5 2 7 8

5 1 6 0 1 3 2 7

A. Which team had the higher median score?

B. Which team had a wider range of scores?



Check It Out! Example 2 The temperature in degrees Celsius for two weeks are given in the stem-and-leaf plot below. Use the data to answer the following questions.

Temperature in Degrees Celsius

Stem Leaves

0 71 92 2 3 6 7 9 3 0 1 1 2 4 5 6



Check It Out! Example 2

A. What was the range of temperatures during the two weeks?


Stem Leaves

0 71 92 2 3 6 7 9 3 0 1 1 2 4 5 6

B. What was the mean temperature during the two weeks?




C. How many days had temperatures above freezing?


Stem Leaves

0 71 92 2 3 6 7 9 3 0 1 1 2 4 5 6

D. What percent of the days had temperatures in the 20’s?



The frequency of a data value is the number of times it occurs. A frequency table shows the frequency of each data value. If the data is divided into intervals, the table shows the frequency of each interval.



A histogram is a bar graph used to display the frequency of data divided into equal intervals. The bars must be of equal width and should touch, but not overlap.



Example 3: Reading a Frequency Table

The numbers of students enrolled in Western Civilization classes at a university are given on the frequency table below. Use the table to answer the following questions.

Number Enrolled

Frequency

1 – 10 1

11 – 20 4

21 – 30 5

31 – 40 2

Enrollment in WesternCivilization Classes


10-2 Frequency and HistogramsExample 3: Reading a Frequency Table

A. How many classes have more than 20 students?

Number Enrolled

Frequency

1 – 10 1

11 – 20 4

21 – 30 5

31 – 40 2

Enrollment in WesternCivilization Classes

B. If a class were selected at random, what is the probability that the class would have more than 30 students?



The number of days of Maria’s last 15 vacations are shown on the frequency table below. Use the table to answer the following questions.


Interval Frequency

4 – 6 5

7 – 9 4

10 – 12 4

13 – 15 2

Number of Vacation Days



A. How many more vacations lasted 4 to 6 days than lasted 13 to 15 days?


Interval Frequency

4 – 6 5

7 – 9 4

10 – 12 4

13 – 15 2

Number of Vacation Days

B. What percent of the vacations lasted 7 to 9 days?



Example 4The histogram shows the frequencies for various enrollments in Civilizations classes. Use the histogram to answer the following questions.

A. How many classes have between 11 and 30 students?



Example 4The histogram shows the frequencies for various enrollments in Civilizations classes. Use the histogram to answer the following questions.

B. If you were enrolled into a class at random, what is the probability of there being at most 10 students?




The histogram shows the frequency for the different heights of Black Cherry Trees in an orchard. Use the histogram to answer the following questions.

A. For which range of heights are there the fewest in the orchard?

B. If a tree were selected at random, you would expect its height to be in what range?



Cumulative frequency shows the frequency of all data values less than or equal to a given value. You could just count the number of values, but if the data set has many values, you might lose track. Recording the data in a cumulative frequency table can help you keep track of the data values as you count.



Example 4: Making a Cumulative Frequency Table

The weights (in ounces) of packages of cheddar cheese are given on the table below. Use the table to answer the following questions.

Weight (oz)

Frequency CumulativeFrequency

18-20 6 6

21-23 2 8

24-26 5 13

27-29 3 16

Cheddar Cheese



Example 5: Reading a Cumulative Frequency Table

A. How many packages weighed less than 24 ounces?

Weight (oz)

Frequency CumulativeFrequency

18-20 6 6

21-23 2 8

24-26 5 13

27-29 3 16

Cheddar Cheese

B. What percent of packages weighed less than 27 ounces?




The number of vowels in each sentence of a short essay are shown on the cumulative frequency table below. Fill in the missing values.

Number Frequency CumulativeFrequency

28-31 2 2

32-35 ___ 9

36-39 5 ___

40-43 3 ___

Vowels in Sentences

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