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The data can be put in order from lowest to highest: 20, 68, 80, 92.
The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20.
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:
[link] lists the different data values in ascending order and their frequencies.
DATA VALUE | FREQUENCY |
---|---|
2 | 3 |
3 | 5 |
4 | 3 |
5 | 6 |
6 | 2 |
7 | 1 |
A frequency is the number of times a value of the data occurs. According to [link] , there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. 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.
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 values in the relative frequency column of [link] 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 tothe relative frequency for the current row, as shown in [link] .
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 0.30 | 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.
[link] represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
HEIGHTS
(INCHES) |
FREQUENCY | RELATIVE
FREQUENCY |
CUMULATIVE
RELATIVE FREQUENCY |
---|---|---|---|
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 |
Total = 100 | Total = 1.00 |
The data in this table have been grouped into the following intervals:
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