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You likely already know about frequency, which refers to how often a data value occurs. Cumulative means the accumulation of everything that has occurred up to a certain point. Therefore, cumulative frequency is the collected frequency of data points.
EXAMPLE
If a teacher says that a test is cumulative, that means that it's going to cover everything that you've learned that year, up to the point of the test.In this context, cumulative frequencies involve separating the data into bins and determining how many observations fall within or below that bin.
EXAMPLE
This is the distribution of temperatures by 10's for Chanhassen, Minnesota in the year 2009. Three days were between -10℉ and -1℉, eight days that were between 0℉ and 9℉ for the high temperature, and so forth.Temperature | Frequency |
---|---|
-10 - -1 | 3 |
0 - 9 | 8 |
10 - 19 | 25 |
20 - 29 | 39 |
30 - 39 | 30 |
40 - 49 | 51 |
50 - 59 | 46 |
60 - 69 | 39 |
70 - 79 | 80 |
80 - 89 | 40 |
90 - 99 | 4 |
Temperature | Frequency | Cumulative Freq |
---|---|---|
-10 - -1 | 3 | 3 |
0 - 9 | 8 | 11 |
10 - 19 | 25 | 36 |
20 - 29 | 39 | 75 |
30 - 39 | 30 | 105 |
40 - 49 | 51 | 156 |
50 - 59 | 46 | 202 |
60 - 69 | 39 | 241 |
70 - 79 | 80 | 321 |
80 - 89 | 40 | 361 |
90 - 99 | 4 | 365 |
Sometimes it is useful to consider relative cumulative frequencies, which is the percent of observations that fall in or below a certain bin.
You may have encountered relative frequency before, but not relative cumulative frequency. Fortunately, it's calculated the same way as relative frequency. To determine the relative cumulative frequency, divide each value by the total number of values.
In the above example, we are considering a full year or 365 days. So we will divide each cumulative frequency by 365 to get the relative cumulative frequency.
In the first bin, there were 3 out of 365 values that fell in this category. This means that 0.008 of the data fell in or below this bucket. Dividing 11 by 365 gives you about 0.03. Continuing on the rest of the chart, we get these values.
Temperature | Frequency | Cumulative Freq | Rel. Cumulative Freq |
---|---|---|---|
-10 - -1 | 3 | 3 | /365 = 0.008 |
0 - 9 | 8 | 11 | /365 = 0.030 |
10 - 19 | 25 | 36 | /365 = 0.099 |
20 - 29 | 39 | 75 | /365 = 0.205 |
30 - 39 | 30 | 105 | /365 = 0.288 |
40 - 49 | 51 | 156 | /365 = 0.427 |
50 - 59 | 46 | 202 | /365 = 0.533 |
60 - 69 | 39 | 241 | /365 = 0.660 |
70 - 79 | 80 | 321 | /365 = 0.879 |
80 - 89 | 40 | 361 | /365 = 0.989 |
90 - 99 | 4 | 365 | /365 = 1.000 |
In the previous chart, you may notice that the final value of 1.000 means that 100% of the values, or all 365 days, fell at or below this bin. Graphically, this information can be presented in something called an ogive. It's also called a relative cumulative frequency graph, or sometimes a percentile graph. It's a line chart that uses these bins and the relative cumulative frequencies to show how many values were at or below these bins.
Temperature | Rel. Cum. Freq | |
---|---|---|
-10 - -1 | 0.008 | |
0 - 9 | 0.030 | |
10 - 19 | 0.099 | |
20 - 29 | 0.205 | |
30 - 39 | 0.288 | |
40 - 49 | 0.427 | |
50 - 59 | 0.533 | |
60 - 69 | 0.660 | |
70 - 79 | 0.879 | |
80 - 89 | 0.989 | |
90 - 99 | 1.000 |
Use the left-hand edge of the bin, because by the time you’ve gotten to negative 10 degrees going left to right on this number line, you haven't encountered any of the days of the year yet. However, once you get to zero degrees, you’ve encountered three of the days, which is a certain amount of relative cumulative frequency. By the time you get to 100 degrees, you will have encountered every single day. Every day will have been at some point in or below that bin.
Source: THIS TUTORIAL WAS AUTHORED BY JONATHAN OSTERS FOR SOPHIA LEARNING. PLEASE SEE OUR TERMS OF USE.