# 1.10 Sampling and data: frequency, relative frequency, and cumulative

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This module introduces the concepts of frequency, relative frequency, and cumulative relative frequency, and the relationship between these measures. Students will have the opportunity to interpret data through the sample problems provided.Labs modified to include minitabs.

Twenty students were asked how many hours they worked per day. Their responses, in hours, are listed below:

• 5
• 6
• 3
• 3
• 2
• 4
• 7
• 5
• 2
• 3
• 5
• 6
• 5
• 4
• 4
• 3
• 5
• 2
• 5
• 3

Below is a frequency table listing the different data values in ascending order and their frequencies.

Frequency table of student work hours
DATA VALUE FREQUENCY
2 3
3 5
4 3
5 6
6 2
7 1

A frequency is the number of times a given datum occurs in a data set. According to the table above, there are three students who work 2 hours, five students who work 3 hours, etc. The total of the frequency column, 20, represents the total number of students included in the sample.

A relative frequency is the fraction of times an answer occurs. To find the relative frequencies, divide each frequency by the total number of students in the sample - in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Frequency table of student work hours w/ realative frequency
DATA VALUE FREQUENCY RELATIVE FREQUENCY
2 3 $\frac{3}{20}$ or 0.15
3 5 $\frac{5}{20}$ or 0.25
4 3 $\frac{3}{20}$ or 0.15
5 6 $\frac{6}{20}$ or 0.30
6 2 $\frac{2}{20}$ or 0.10
7 1 $\frac{1}{20}$ or 0.05

The sum of the relative frequency column is $\frac{20}{20}$ , or 1.

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.

Frequency table of student work hours w/ relative and cumulative frequency
DATA VALUE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY
2 3 $\frac{3}{20}$ or 0.15 0.15
3 5 $\frac{5}{20}$ or 0.25 0.15 + 0.25 = 0.40
4 3 $\frac{3}{20}$ or 0.15 0.40 + 0.15 = 0.55
5 6 $\frac{6}{20}$ or 0.10 0.55 + 0.30 = 0.85
6 2 $\frac{2}{20}$ or 0.10 0.85 + 0.10 = 0.95
7 1 $\frac{1}{20}$ or 0.05 0.95 + 0.05 = 1.00

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

Because of rounding, the relative frequency column may not always sum to one and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.

The following table represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

Frequency table of soccer player height
HEIGHTS (INCHES) FREQUENCY OF STUDENTS RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY
Total = 100 Total = 1.00
59.95 - 61.95 5 $\frac{5}{100}$ = 0.05 0.05
61.95 - 63.95 3 $\frac{3}{100}$ = 0.03 0.05 + 0.03 = 0.08
63.95 - 65.95 15 $\frac{15}{100}$ = 0.15 0.08 + 0.15 = 0.23
65.95 - 67.95 40 $\frac{40}{100}$ = 0.40 0.23 + 0.40 = 0.63
67.95 - 69.95 17 $\frac{17}{100}$ = 0.17 0.63 + 0.17 = 0.80
69.95 - 71.95 12 $\frac{12}{100}$ = 0.12 0.80 + 0.12 = 0.92
71.95 - 73.95 7 $\frac{7}{100}$ = 0.07 0.92 + 0.07 = 0.99
73.95 - 75.95 1 $\frac{1}{100}$ = 0.01 0.99 + 0.01 = 1.00

The data in this table has been grouped into the following intervals:

• 59.95 - 61.95 inches
• 61.95 - 63.95 inches
• 63.95 - 65.95 inches
• 65.95 - 67.95 inches
• 67.95 - 69.95 inches
• 69.95 - 71.95 inches
• 71.95 - 73.95 inches
• 73.95 - 75.95 inches

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