Frequency distribution tables

Frequency Distribution Tables

Frequency is how often something occurs. For example; Amina plays netball twice on Monday, once on Tuesday and thrice on Wednesday. Twice, once and thrice are frequencies.

By counting frequencies we can make Frequency Distribution table. Frequency Distribution Tables from Raw Data

For example; Sam's team has scored the following goals in recent games.

2, 3, 1, 2, 1, 3, 2, 3, 4, 5, 4, 2, 2, 3.

How to make a frequency distribution table

Put the number in order i.e. 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5

Write how often a certain number occurs. This is called tallying

1. how often 1 occurs? (2 times)
2. how often 2 occurs? (5 times)
3. how often 3 occurs? (4 times)
4. how often 4 occurs? (2 times)
5. how often 5 occurs? (1 time)

Then, wrote them down on a table as a Frequency distribution table.

Scores	Frequency
1	2
2	5
3	4
4	2
5	1

From the table we can see how many goals happen often, and how many goals they scored once and so on.

Interpretation of Frequency Distribution Table from Raw Data

Grouped Distribution Table

This is very useful when the scores have many different values. For example; Alex measured the lengths of leaves on the Oak tree (to the nearest cm)

9, 16, 13, 7, 8, 4, 18, 10, 17, 18, 9, 12, 5, 9, 9, 16, 1, 8, 17, 1, 10, 5, 9, 11, 15, 6, 14, 9, 1, 12, 5, 16, 4, 16, 8, 15, 14, 17.

How to make a grouped distribution table

Step 1: Put the numbers in order. 1, 1, 1, 4, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18,

Step 2: Find the smallest and the largest values in your data and calculate the range.

The smallest (minimum) value is 1 cm

The largest (maximum) value is 18 cm

The range is 18 cm – 1 cm = 17 cm

Step 3: Find the size of each group. Calculate an approximate size of the group by dividing the range by how many groups you would like. then, round that group size up to some simple value like 4 instead of 4.25 and so on.

Let us say we want 5 groups. Divide the range by 5 i.e. 17/5 = 3.4. then round up to 4

Step 4: Pick a Starting value that is less than or equal to the smallest value. Try to make it a multiple of a group size if you can. In our case a start value of 0 make the most sense.

Step 5: Calculate the list of groups (we must go up to or past the largest value).

In our case, starting at 0 and with a group size of 4 we get 0, 4, 8, 12, 16. Write down the groups. Include the end value of each group. (must be less than the next group):

The largest group goes up to 19 which is greater than the maximum value. This is good.

Step 6: Tally to find the frequencies in each group and then do a total as well

Length (cm)	Frequency
0 - 3	3
4 - 7	7
8 - 11	12
12 - 15	7
16 - 19	9
Total	38

Done!

Upper and Lower values

Referring our example; even though Alex measured in whole numbers, the data is continuous. For instance 3 cm means the actual value could have been any were between 2.5 cm to 3.5 cm. Alex just rounded numbers to whole numbers. And 0 means the actual value have been any where between -0.5 cm to 0.5 cm. but we can't say length is negative. 3.5 cm is called upper real limit or upper boundary while –0.5 cm is called lower real limit or lower boundary. But since we don't have negative length we will just use 0. So regarding our example the lower real limit is 0.

The limits that we used to group the data are called limits. For example; in a group of 0 – 3, 0 is called lower limit and 3 is called upper limit.

See an illustration below to differentiate between Real limits and limits.

Class size is the difference between the upper real limit and the lower real limit i.e. class size = upper real limit – lower real limit

We use the symbol N (capital N) to represent the total number of frequencies.

Class Mark of a class Interval

This is a central (middle) value of a class interval. It is a value which is half way between the class limits. It is sometimes called mid-point of a class interval. Class mark is obtained by dividing the sum of the upper and lower class limits by 2. i.e.

Class mark

$\text{Class Mark} = \frac{\text{upper class limit} + \text{lower class limit}}{2}$

Referring to our example class marks for the class intervals are;

$\frac{0+3}{2} = 1.5$

$\frac{4+7}{2} = 5.5$

Interpretation of Frequency Distribution Tables

Example 5 interpretation of frequency distribution data:

A survey was conducted to determine the number of people in cars during rush hour. The results are shown in the frequency diagram below.

Total number of cars in the survey:

6 + 3 + 5 + 1 = 15

There are 6 cars with one person in, 3 cars with two people, 5 cars with three people, and 1 car with four people.

the most likely number of people in a car:

Cars in the survey are most likely to have 1 person in them as this is the tallest bar - 6 of the cars in the survey had one occupant.