A set of data can be described with a frequency distribution. There are two types of frequency distributions: Grouped and Ungrouped (“Psychological statistics. Frequency Distributions”, N.p.).
The following rules must be completed in order to create an ungrouped frequency distribution (“Psychological statistics. Frequency Distributions”, N.p.):
- Set the values of data, which are called scores, in the column starting from the lowest value to the highest or vice versa;
- Create the second column with the frequency of each data occurrence. This column is known as the tally of the scores;
- Create the third column, where the relative frequency of each score will be inserted. The relative frequency can be obtained as follows: fr=f/N, there f is the frequency of each score (from the second column) and N is the total number of scores. In order to check the correctness of calculations, the sum of fr should be calculated and should be equal to 1.
- The next column, where the relative frequency will be performed in percentages, is to be created.
- In the next column, known as the cumulative frequency column, the cumulative frequency for each score should be estimated. Calculation of the cumulative frequency, should be started from the lowest value of score, for which the cumulative frequency equals the value of frequency from the second column. The further calculations are to be performed for each score in a sequence from lowest to highest and the cumulative frequency for each next score equals to the sum of the cumulative frequency of the previous score and frequency of this score (from the second column). The cumulative frequency of the highest score should be equal to the total number of scores.
- The next column is called “cumulative proportion” and the values of its column are obtained as a ratio of cumulative frequency for each score and the total number of scores.
- The last column is the cumulative percent, where the cumulative proportion is presented as percentages.
Problem 1. Create an Ungrouped Frequency Distribution table with the data from the survey, accomplished among the students of university, which answered the question of how many books they read per year. The answers are presented below:
| 7 | 3 | 0 | 9 | 8 | 5 | 4 | 4 | 5 | 6 | 6 | 3 | 3 | 4 | 5 | 1 | 0 | 3 |
Solution:
The next steps were accomplished:
- First column where all different answers were placed from higher to lower value was created;
- Second column, where quantity of same answers in the survey was put for each score, was created;
- Third column was obtained by dividing values of the second column on the number of answers;
- Fourth column was obtained by multiplying the values from the third column by 100%;
- The values for the fifth column were obtained starting from the lowest score (0), for which Cf=f, and for other scores Cf was obtained as the sum of Cf for the previous score and the f from the second column for current score.
- Cp for the sixth column was calculated as a ratio of Cf and N;
- the last column was obtained by multiplying the sixth column on 100%.
See results of calculations in the table below.
| X | f | fr | % | Cf | Cp | C% |
|---|---|---|---|---|---|---|
| 9 | 1 | 1/18=0,06 | 6 | 18=N | 18/18=1 | 100 |
| 8 | 1 | 0,06 | 6 | 17 | 17/18=0,94 | 94 |
| 7 | 1 | 0,05 | 5 | 16 | 0,88 | 88 |
| 6 | 2 | 0,11 | 11 | 15 | 0,83 | 83 |
| 5 | 3 | 0,16 | 16 | 13 | 0,72 | 72 |
| 4 | 3 | 0,17 | 17 | 10 | 0,56 | 56 |
| 3 | 4 | 0,22 | 22 | 7 | 0,39 | 39 |
| 2 | 0 | 0 | 0 | 3 | 0,17 | 17 |
| 1 | 1 | 0,06 | 6 | 3 | 0,17 | 17 |
| 0 | 2 | 0,11 | 11 | 2 | 0,11 | 11 |
| N=18 | Σ=1 | Σ%=100 | ||||
Problem 2. Create an Ungrouped Frequency Distribution table with the data from the survey concerning the age of the people, which attend the gym. The results of a survey are presented below.
| 25 | 30 | 27 | 32 | 22 | 25 | 27 | 24 | 25 | 26 | 26 | 23 | 23 | 24 | 25 | 21 | 20 | 30 |
| 31 | 25 | 30 | 32 | 21 | 21 | 24 | 25 | 26 | 28 | 30 | 29 | 32 | 19 | 20 | 21 | 25 | 25 |
Solution
The same steps as for the previous problem were accomplished. Results are performed in the Ungrouped Frequency Distribution table below.
| X | f | fr | % | Cf | Cp | C% |
|---|---|---|---|---|---|---|
| 19 | 1 | 0,028 | 2,8 | 36=N | 1 | 100 |
| 20 | 2 | 0,056 | 5,6 | 35 | 0,972 | 97,2 |
| 21 | 4 | 0,111 | 11,1 | 33 | 0,917 | 91,7 |
| 22 | 1 | 0,028 | 2,8 | 29 | 0,806 | 80,6 |
| 23 | 2 | 0,056 | 5,6 | 28 | 0,778 | 77,8 |
| 24 | 3 | 0,083 | 8,3 | 26 | 0,722 | 72,2 |
| 25 | 8 | 0,222 | 22,2 | 23 | 0,639 | 63,9 |
| 26 | 3 | 0,083 | 8,3 | 15 | 0,417 | 41,7 |
| 27 | 2 | 0,056 | 5,6 | 12 | 0,333 | 33,3 |
| 28 | 1 | 0,028 | 2,8 | 10 | 0,278 | 27,8 |
| 29 | 1 | 0,028 | 2,8 | 9 | 0,25 | 25 |
| 30 | 4 | 0,111 | 11,1 | 8 | 0,222 | 22,2 |
| 31 | 1 | 0,028 | 2,8 | 4 | 0,111 | 11,1 |
| 32 | 3 | 0,081 | 8,1 | 3 | 0,083 | 8,3 |
| N=36 | Σ=1 | Σ%=100 | ||||
References
“Psychological Statistics. Frequency Distributions.” Psych. Statistics: Frequency Distributions. Web. 13 May 2016. <https://www4.uwsp.edu/psych/stat/3/frqdist.htm>
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