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Teacher professional development and classroom resources across the curriculum  MENU          Session 3, Part C:
Relative and Cumulative Frequencies (30 minutes)

In This Part: Relative Frequencies | Cumulative Frequencies
Relative Cumulative Frequencies

The frequency histogram and grouped frequency table for the 52 time estimates contain similar information to the stem and leaf plot, but they don't indicate each person's actual estimate. The height of each bar in the histogram indicates the frequency of the corresponding interval of estimates on the horizontal axis. Note 4

As with the stem and leaf plot, the frequency histogram can be an awkward graph for large data sets, since the vertical axis corresponds to the frequency of each interval of values. For large data sets, some intervals may have many values and a high frequency. Consequently, the vertical axis would have to be scaled according to the largest frequency.

An alternative is to use relative frequencies to describe how many values are in each interval relative to the total number of values. For most purposes, relative frequencies are more useful than absolute frequencies; for example, the statement "17 of the 52 estimates are in the interval 50 to < 60" is more useful than the statement "17 estimates are in the interval 50 to < 60."

The relative frequency for the interval 50 to < 60 is 17/52, which you can also write in decimal form as .327 (rounded to three digits). Multiplying by 100 gives you the percentage, 32.7%. This means that 32.7% of the estimates are in the interval 50 to < 60.

Here is what you get for the rest of the data:  Interval Frequency Relative Frequency Fraction Decimal %  30 to < 40 4 4/52 .077 7.7 40 to < 50 1 1/52 .019 1.9 50 to < 60 17 17/52 .327 32.7 60 to < 70 18 18/52 .346 34.6 70 to < 80 7 7/52 .135 13.5 80 to < 90 4 4/52 .077 7.7 90 to < 100 1 1/52 .019 1.9 Notice that the relative frequencies expressed as fractions and decimals add up to 1 and that the percentages add up to 100%.  Problem C1 Use only the relative frequencies from the table to answer the questions below. Give your answers as percentages, to the nearest 10th of a percent, or explain why the answer cannot be found from the table.

 a. What percentage of the responses are in the 70s and below? b. What percentage of the responses are 80 or higher? c. What percentage of the responses are in the 50s and below? d. What percentage of the responses are 60 or higher? e. What percentage of the responses are less than 100? f. What percentage of the responses are at least 40 but below 70? g. What percentage of the responses are 65 or greater? h. What percentage of the responses are less than 35? i. What percentage of the responses are equal to 60?  To determine whether a question can be answered, decide whether you have all the information you would need to answer it.   Close Tip To determine whether a question can be answered, decide whether you have all the information you would need to answer it.      Problem C2 For questions (g) and (h) in Problem C1, use the table to come up with an estimated percentage.  One assumption you might make is that each interval is divided evenly. So if the interval states that 15.4% of the estimates are between 80 and 90, you might assume that half of these (7.7%) are between 80 and 85 and half are between 85 and 90.   Close Tip   The relative frequency histogram looks similar to the frequency histogram; the only differences are that the labels along the vertical axis represent percentages, and the height of each bar now represents the relative frequency expressed as a percentage (or proportion) for the corresponding interval of values.    Session 3: Index | Notes | Solutions | Video

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