In statistics, absolute frequency is a number that expresses the number of values in a data set. The cumulative frequency is not the same as the absolute frequency. Cumulative frequency is the final sum (or most recent sum) of all frequencies to some extent in a data set. The explanation may sound complicated, but don't worry: this topic will be easier to understand if you provide paper and pen and work on the sample problems described in this article.
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Part 1 of 2: Calculating Ordinary Cumulative Frequency
Step 1. Sort the values in the data set
A "data set" is a group of numbers that describes the state of a thing. Sort the values, which are in the data set, from smallest to largest.
Example: You collect data on the number of books read by each student in the past month. The data you get, after sorted from smallest to largest, are: 3, 3, 5, 6, 6, 6, 8
Step 2. Calculate the absolute frequency of each value
The frequency of a value is the number of values it has in the data set (this frequency may be called the “absolute frequency” so as not to be confused with the cumulative frequency). The easiest way to calculate frequency is to create a table. Write “Value” (or what that value measures) in the top row of the first column. Write “Frequency” in the top row of the second column. Fill in the table according to the data set.
- Example: Write "Number of Books" in the top row of the first column. Write “Frequency” in the top row of the second column.
- On the second line, write the first value, which is “3”, under “Number of Books”.
- Count the number of 3 in the data set. Since there are two 3's, write "2" under "Frequency" (on the second line).
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Insert all values into the table:
- 3 | F = 2
- 5 | F = 1
- 6 | F = 3
- 8 | F = 1
Step 3. Calculate the cumulative frequency of the first value
Cumulative frequency is the answer to the question "how many times does this value or a smaller value appear in the data set?" The cumulative frequency calculation must start from the smallest value. Since no value is smaller than the smallest value, the cumulative frequency of that value is equal to its absolute frequency.
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Example: The smallest value in the data set is 3. The number of students who read 3 books is 2 people. No student reads less than 3 books. So, the cumulative frequency of the first value is 2. Write “2” next to the frequency of the first value, in the table:
3 | F = 2 | Fkum=2
Step 4. Calculate the cumulative frequency of the next value in the table
We've just counted the number of times the smallest value appears in the data set. To calculate the cumulative frequency of the next value, add up the absolute frequency of this value with the cumulative frequency of the previous value.
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Example:
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3 | F = 2 | Fkum =
Step 2.
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5 | F =
Step 1. | Fkum
Step 2
Step 1. = 3
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Step 5. Repeat the procedure to calculate the cumulative frequency of all values
Calculate the cumulative frequency of each subsequent value: add up the absolute frequency of a value with the cumulative frequency of the previous value.
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Example:
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3 | F = 2 | Fkum =
Step 2.
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5 | F = 1 | Fkum = 2 + 1 =
Step 3.
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6 | F = 3 | Fkum = 3 + 3 =
Step 6.
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8 | F = 1 | Fkum = 6 + 1 =
Step 7.
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Step 6. Check the answers
After finishing calculating the cumulative frequency of the largest value, the number of each value has been added up. The final cumulative frequency is equal to the number of values in the data set. Check it using one of the following methods:
- Add up the absolute frequencies of all values: 2 + 1 + 3 + 1 = 7. So, “7” is the final cumulative frequency.
- Count the number of values in the data set. The data set in the example is 3, 3, 5, 6, 6, 6, 8. There are 7 values. So, “7” is the final cumulative frequency.
Part 2 of 2: Doing More Complicated Problems
Step 1. Learn about discrete and continuous data
Discrete data in the form of units that can be calculated and each unit cannot be a fraction. Continuous data describes something that cannot be calculated and the measurement results can be in the form of fractions/decimals with whatever units are used. Example:
- The number of dogs is discrete data. The number of dogs cannot be “half a dog”.
- Snow depth is continuous data. The snow depth increases gradually, not one unit at a time. If measured in centimeters, the snow depth might be 142.2 cm.
Step 2. Group continuous data into ranges
Continuous data sets often consist of many unique values. Using the method described above, the final table obtained may be very long and difficult to understand. Therefore, create a specific range of values on each row. The distance between each range must be the same (eg 0-10, 11–20, 21–30, and so on), regardless of how many values are in each range. The following is an example of a continuous data set written in tabular form:
- Data set: 233, 259, 277, 278, 289, 301, 303
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Table (first column is value, second column is frequency, third column is cumulative frequency):
- 200–250 | 1 | 1
- 251–300 | 4 | 1 + 4 = 5
- 301–350 | 2 | 5 + 2 = 7
Step 3. Create a line graph
After calculating the cumulative frequency, prepare graph paper. Draw a line graph with the x-axis as the values in the data set and the y-axis as the cumulative frequency. This method makes further calculations easier.
- Example: if the data set is 1-8, create an x-axis with eight marks. At each value on the x-axis, draw a point according to the value on the y-axis, according to the cumulative frequency of that value. Connect pairs of adjacent dots with lines.
- If a specific value is not present in the data set, the absolute frequency is 0. Adding 0 to the last cumulative frequency does not change the value. So, draw a point at the same y-value as the last value.
- Because the cumulative frequency is directly proportional to the values in the data set, the line graph always increases to the top right. If the line graph is descending, you may see an absolute frequency column instead of a cumulative frequency.
Step 4. Find the median value using a line graph
The median is the value that is right in the middle of the data set. Half the values in the data set are above the median, and the remaining half are below the median. Here's how to find the median value on a line graph:
- Notice the last dot at the far right of the line graph. The y-value of the point is the total cumulative frequency, i.e. the number of values in the data set. For example, the total cumulative frequency of a data set is 16.
- Divide the total cumulative frequency by 2, then find the location of the divided number on the y-axis. In the example, 16 divided by 2 equals 8. Find the “8” on the y-axis.
- Find the point on the line graph that is parallel to the y-value. With your finger, draw a straight line to the side from the “8” position on the y-axis until it touches the line graph. The point touched by the finger in the line graph has crossed half the data set.
- Find the x-value of the point. With your finger, draw a straight line down from the point on the line graph until it touches the x-axis. The point touched by the finger on the x-axis is the median value of the data set. For example, if the median value found is 65, half of the data set is below 65 and the remaining half is above 65.
Step 5. Find the quartile value using a line graph
Quartile values divide the data set into four parts. The method of finding the quartile value is almost the same as the method of finding the median value; just a way of finding a different y value:
- To find the lower quartile y value, divide the total cumulative frequency by 4. The x value that coordinates with the y value is the lower quartile value. A quarter of the data set is below the lower quartile value.
- To find the upper quartile y value, multiply the total cumulative frequency by. The value of x that coordinates with the value of y is the upper quartile value. Three-quarters of the data set is below the upper quartile value and the remaining quarter is above the upper quartile value. of the entire data set.
