Definition of Frequency tables with examples
This article will be about the frequency tables, definition of the tables, how to make one of them with all the parts and examples.
The frequency tables in statistics are used to order the data compiled in a study. The frequency tables are useful to present and analyze the data of a more structured way, so it can be easier to see the data, this tables represent every possible variable and the frequency of each data, and not only the frequency, also other useful types of frequency.
This tables are useful to classify the data compiled in a determine study of a way where we can see various points of view, as it could be the occasions that a variable was obtained, the percentage of each of the variables, and which percentages are before and after a variable.
A frequency table is composed by the following columns: the absolute frequency ni, the cumulative absolute frequency Ni, the relative frequency fi, the cumulative relative frequency Fi, the percentage frequency f% and the cumulative percentage frequency F%. Although this columns are the frequencies in a frequency table, there are sometimes where we just find the columns we need.
| Value | ni | Ni | fi | Fi | f% | F% |
|---|---|---|---|---|---|---|
| 1 | 4 | 4 | 0.5 | 0.5 | 50% | 50% |
| 2 | 4 | 8 | 0.5 | 1 | 50% | 100% |
| Total | 8 | 1 | 100% |
Structure of the frequency tables
To be clear about how to make a frequency table, we are going to use the following example: A group of people was asked about the quantity of glasses of water they drink in one single day, and the results were the following.
{2,3,5,2,7,8,5,4,6,3,6,3,4,6,2,5,8,9,5,2}
Absolute frequency (ni).
The absolute frequency is the sum of the times that a value is repeated. To find the absolute frequency in the previous example first we have to define all the values that were an answer, no matter the times the value was repeated, in this case the answers were: 2,3,4,5,6,7,8 and 9, and this values will be in the first column. After this, we are going to count the number of times that each value was repeated, for example, if we look at the data we can see that the number 2 was repeated 4 times, so the frequency of 2 will be equals to 4, and we are going to do this with every answer. And this will be the second column of the frequency table. At the end of the second column we have to make a total of these frequencies.
| Glasses/day | ni |
|---|---|
| 2 | 4 |
| 3 | 3 |
| 4 | 2 |
| 5 | 4 |
| 6 | 3 |
| 7 | 1 |
| 8 | 2 |
| 9 | 1 |
| Total | 20 |
cumulative absolute frequency (Ni).
This frequency is obtained by adding the absolute frequency with every absolute frequency above it, for example, to calculate the cumulative frequency of the number 3, we would have to sum the absolute frequency of 2 plus the absolute frequency of 3, and when we do this with every row, the last cumulative frequency will have to coincide with the total of absolute frequency.
| Glasses/day | ni | Ni |
|---|---|---|
| 2 | 4 | 4 |
| 3 | 3 | 7 |
| 4 | 2 | 9 |
| 5 | 4 | 13 |
| 6 | 3 | 16 |
| 7 | 1 | 17 |
| 8 | 2 | 19 |
| 9 | 1 | *20 |
| Total | *20 |
Relative frequency (fi).
The relative frequency is obtained by dividing the frequency by the total of the absolute frequencies, this is going to be a number between 0 and 1, and this represents the proportion of a result.
Once we have calculated all the relative frequencies of each class, we have to sum the total of relatives frequencies, and this number will have to be equals to 1 (or something like 0.99…), in case the result is not approximately 1, then the result is wrong.
| Glasses/day | ni | Ni | fi |
|---|---|---|---|
| 2 | 4 | 4 | 0.2 |
| 3 | 3 | 7 | 0.15 |
| 4 | 2 | 9 | 0.1 |
| 5 | 4 | 13 | 0.2 |
| 6 | 3 | 16 | 0.15 |
| 7 | 1 | 17 | 0.05 |
| 8 | 2 | 19 | 0.1 |
| 9 | 1 | 20 | 0.05 |
| Total | 20 | 1 |
Cumulative relative frequency (Fi).
This is obtained by adding all the previous relative frequencies, where the value of the last frequency will have to be equals to 1.
| Glasses/day | ni | Ni | fi | Fi |
|---|---|---|---|---|
| 2 | 4 | 4 | 0.2 | 0.2 |
| 3 | 3 | 7 | 0.15 | 0.35 |
| 4 | 2 | 9 | 0.1 | 0.45 |
| 5 | 4 | 13 | 0.2 | 0.65 |
| 6 | 3 | 16 | 0.15 | 0.8 |
| 7 | 1 | 17 | 0.05 | 0.85 |
| 8 | 2 | 19 | 0.1 | 0.95 |
| 9 | 1 | 20 | 0.05 | 1 |
| Total | 20 | 1 |
Percentage frequency (f%).
This defines the percentage proportion of a range with respect of the others, this is the percentage of the data that is in the interval, the value of this frequency is between 0% and 100%, and it is obtained by multiplying the relative frequency by 100, and when we sum all the percentage frequencies, the result will have to be equals to 100%.
| Glasses/day | ni | Ni | fi | Fi | f% |
|---|---|---|---|---|---|
| 2 | 4 | 4 | 0.2 | 0.2 | 20% |
| 3 | 3 | 7 | 0.15 | 0.35 | 15% |
| 4 | 2 | 9 | 0.1 | 0.45 | 10% |
| 5 | 4 | 13 | 0.2 | 0.65 | 20% |
| 6 | 3 | 16 | 0.15 | 0.8 | 15% |
| 7 | 1 | 17 | 0.05 | 0.85 | 5% |
| 8 | 2 | 19 | 0.1 | 0.95 | 10% |
| 9 | 1 | 20 | 0.05 | 1 | 5% |
| Total | 20 | 1 | 100% |
Accumulative percentage frequency
Is calculated by adding all the previous percentage frequencies, and the last frequency will have to be equals to 100%.
| Glasses /day | ni | Ni | fi | Fi | f% | F% |
|---|---|---|---|---|---|---|
| 2 | 4 | 4 | 0.2 | 0.2 | 20% | 20% |
| 3 | 3 | 7 | 0.15 | 0.35 | 15% | 35% |
| 4 | 2 | 9 | 0.1 | 0.45 | 10% | 45% |
| 5 | 4 | 13 | 0.2 | 0.65 | 20% | 65% |
| 6 | 3 | 16 | 0.15 | 0.8 | 15% | 80% |
| 7 | 1 | 17 | 0.05 | 0.85 | 5% | 85% |
| 8 | 2 | 19 | 0.1 | 0.95 | 10% | 95% |
| 9 | 1 | 20 | 0.05 | 1 | 5% | 100% |
| Total | 20 | 1 | 100% |
Examples of frequency tables
In this examples when we sum the relative and percentage frequency is probably that the result is not exactly to 1 or 100, this will depend on quantity of decimals we are going to use. So it does not mean that the results are wrong.
Example 1: A poll was made where people were asked about the type of animal that was their first per, and the results were:
| Pet | ni |
|---|---|
| dog | 41 |
| cat | 23 |
| turtle | 13 |
| Total | 77 |
Frequency table developed
| Pet | ni | Ni | fi | Fi | f% | F% |
|---|---|---|---|---|---|---|
| Dog | 41 | 41 | 0.53 | 0.53 | 53% | 53% |
| Cat | 23 | 64 | 0.3 | 0.83 | 30% | 83% |
| Turtle | 13 | 77 | 0.17 | 1 | 17% | 100% |
| Total | 77 | 1 | 100% |
With the data structured this way we can arrive to different analyzes, like the 73% of the persons had as a first pet either a cat or a dog, and this is one of many analyses these tables will help us to have.
Example 2: A group of sport men was asked about what was their favorite sport, the answers were: football, basketball, volleyball and swimming, and the answers were:
| Sport | ni |
|---|---|
| Football | 86 |
| Basketball | 57 |
| Volleyball | 30 |
| Swimming | 12 |
| Total | 185 |
Frequency table solved
| Pet | ni | Ni | fi | Fi | f% | F% |
|---|---|---|---|---|---|---|
| Football | 86 | 86 | 0.46 | 0.46 | 46% | 46% |
| Basketball | 57 | 143 | 0.31 | 0.77 | 31% | 77% |
| Volleyball | 30 | 173 | 0.16 | 0.93 | 16% | 93% |
| Swimming | 12 | 185 | 0.06 | 1 | 6% | 100% |
| Total | 185 | 1 | 100% |