Curricular Resources > 1986 Volume V > Unit 3 (86.05.03) > Section 4

The Measurement of Adolescents, II

CONTENTS OF CURRICULUM UNIT 86.05.03

Unit Guide

AN INTRODUCTION TO ELEMENTARY STATISTICS

Lauretta J. Fox

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Frequency Distributions

Groups of data have little value until they have been placed in some kind of order. Usually measurements are arranged in ascending or descending order. Such a group is an array or distribution . A frequency distribution is a table in which measurements are tallied and the frequency or total number of times that each item occurs is recorded.

Example 1 :

The frequency distribution below shows data obtained in a survey asking a group of people to name their favorite among several kinds of cars. Use the table to answer the following questions:

a.) How many people are included in the sample?
b.) What percent of the people surveyed preferred Chevrolets?
c.) What is the ratio of people who prefer Oldsmobiles to those who prefer Buicks?
d.) If the number of Subarus were increased by three, what would the percent of increase be?

(figure available in print form)

Solution:

a.) 25 + 18 + 15 + 12 + 10 = 80 80 people are included in the survey.
b.) 25 Ö 80 = .3125 = 31.25% 31.25% of the people surveyed preferred Chevrolets.
c.) 15:10 = 3:2 The ratio of people who prefer Oldsmobiles to those who prefer Buicks is 3:2.
d.) 3:18=x:100 1:6 =x:100 6x=100 x =16 2/3 The percent of increase is 16 2/3%.

When the number of measurements in a survey is large, or when the range, that is, the difference between the highest and lowest measurements in the survey, is great, it is usually more efficient to arrange the data in intervals and show the number of items within each group. The number of intervals used in a frequency distribution may vary. However, it has been found that ten to twenty intervals are most practical.

The following steps may be used to set up a frequency distribution:

1.) Select an appropriate number of intervals for the given data.
2.) Find the difference between the highest and lowest measurements in the data. Add one to the result end divide the sum by the number of intervals. If the quotient is not an integer, round it to the nearest odd integer. This will be the size or width of each interval and will be designated by the symbol w.
3.) The lowest number in the bottom interval will be the lowest measurement in the given data. Add (w-1) to this measurement to obtain the highest number in the bottom interval. The next interval begins at the integer following the highest number in the bottom group. Continue in this manner for each successive higher interval until every measurement has been placed in its proper group.
4.) After the intervals have been established, a tally mark is placed by the interval for each measurement in the group. The frequency, or number of measurements in each interval, is indicated with a numeral.

Example 2 :

Make a frequency distribution of the following scores obtained by 40 students on a mathematics test.

86	82	56	73	87	89	72	86	88	76
72	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

Solution :

Use ten intervals.

Highest Score—Lowest Score = 98Ð56 = 42 (42 + 1) = 10 = 43 $dv$ 10 = 4.3 Round to 5. The size of each interval is 5.

Scores	Tall y	Frequency
96Ð100	111	3
91Ð95	1111	4
86Ð90	111	8
81Ð85	11	7
76Ð80	1	6
71Ð75	1111	5
66Ð70	111	3
61Ð65	11	2
56 Ð60	11	2

Although it is not necessary, it is often helpful for use in further analysis to have additional information in a frequency distribution. This additional information may include the midpoint of each interval, the percentage of the numbers in the frequency column relative to the total frequencies, the cumulative frequency of successive summation of entries in the frequency column, and the percentage of the cumulative frequency.

Example 3 :

In the frequency distribution for example 2 find (a) the midpoint of each interval; (b) the percentage of each frequency relative to the total frequencies; (c) the cumulative frequency; and (d) the percentage of cumulative frequency relative to the total frequencies.

Solution :

(a) Since the width of each interval is 5, the third score is the midpoint of the interval. For example, the lowest interval contains the scores 56, 57, 58, 59, 60. 58 is the midpoint of this interval.
(b) To find the percentage of each frequency divide the frequency by the total number of measurements and change the resulting decimal to a percent. The frequency of the lowest interval is 2. The total number of measurements is 40. 2 $dv$ 40 = .05 = 5%
(c) The cumulative frequency at any interval may be obtained by successively adding the frequencies of all the groups from the lowest interval up to and including the given interval. The cumulative frequency of the interval 76-80 is 2+ 2+ 3+ 5+ 6 =18.
(d) To obtain the percentage of cumulative frequency relative to the total of the frequencies, divide the cumulative frequency by the total number of measurements. Change the resulting decimal to a percent. The percentage of the cumulative frequency in the interval 76-80 is 18 $dv$ 40 = .45= 45%. This figure may also be found by adding the percentage of frequency of all groups from the lowest up to and including the given interval.

			% of	Cumulative	% of
Scores	Midpoint	Frequency	Frequency	Frequency	Cumulative
					Frequency
99-100	98	3	7.5	40	100.0
9195	93	4	10.0	37	92.5
8690	88	8	20.0	33	82.5
8185	83	7	17.5	25	62.5
7680	78	6	15.0	18	45.0
7175	73	5	12.5	12	30.0
6670	68	3	7.5	7	17.5
6165	63	2	5.0	4	10.0
5660	58	2	5.0	2	5.0

Exercises :

1.) Ask the students in each of your classes which of the following colors they prefer—red, blue, yellow, green, brown, or purple. Construct a frequency distribution to display the results of your survey
a.) How many people are included in the sample?

b.) What percent of the people surveyed prefer yellow? red? purple?

c.) What is the ratio of people who prefer green to those who prefer blue?

d.) What is the most popular color?

e.) What is the least popular color?

f.) If the number of people who prefer red were decreased by 2, what would be the percent of decrease?

2.) Tally the following scores in a frequency distribution. Do not use grouping.

84	98	92	88	91	91	85	80	84	93
92	80	91	84	87	85	84	80	87	95

3.) Make a frequency distribution of the following scores obtained by a basketball team.

72	104	95	93	96	76	105	100
88	62	79	78	87	78	89	81
110	68	96	106	80	87	86	84
102	84	96	88	82	83	92	87
87	85	108	90	94	98	78	80

a.) Use ten intervals and display the midpoint of each interval.

b.) Calculate the percentage of frequency of each interval.

c.) Find the cumulative frequency for each interval.

d.) Calculate the percentage of each cumulative frequency relative to the total of the frequencies.

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86	82	56	73	87	89	72	86	88	76
72	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

72	104	95	93	96	76	105	100
88	62	79	78	87	78	89	81
110	68	96	106	80	87	86	84
102	84	96	88	82	83	92	87
87	85	108	90	94	98	78	80

86	82	56	73	87	89	72	86	88	76
72	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

72	104	95	93	96	76	105	100
88	62	79	78	87	78	89	81
110	68	96	106	80	87	86	84
102	84	96	88	82	83	92	87
87	85	108	90	94	98	78	80

86	82	56	73	87	89	72	86	88	76
72	69	84	85	62	97	70	78	84	93
70	60	91	76	83	94	65	72	92	81
98	78	88	76	96	89	90	83	74	80

72	104	95	93	96	76	105	100
88	62	79	78	87	78	89	81
110	68	96	106	80	87	86	84
102	84	96	88	82	83	92	87
87	85	108	90	94	98	78	80