Frequency Distribution Formula: Meaning and Steps

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The frequency distribution formula and table are typically powerful tools in statistical mathematics for determining the correct number of observations and their distribution within the given integrals through the use of the graphical or tabular display. The entire article is based on the frequency distribution formula and the formula for determining group data.

Table of Content

What is Frequency Distribution?
Formula of Frequency Distribution
Steps of Frequency Distribution Construction
Things to Remember
Sample Questions

Key Takeaways: Frequency distribution formula, Frequency, statistics, data, table, distribution

What is Frequency Distribution?

A frequency distribution table is a method of organising data in order to make it more meaningful. A frequency distribution table is a chart that summarises all data into two columns: variables/categories and frequency. It consists of two or three columns. Depending on the size of the data set, the first column will typically list all of the outcomes as individual values or in the form of class intervals. The tally marks for each outcome are listed in the second column. The frequency of each outcome is listed in the third column. The second column is also optional.

The information gathered is referred to as data. Once the data has been collected, it must be represented in a meaningful way so that it can be easily understood. One method for organising data is to use a frequency distribution table.

Also Read:

Mean of Grouped Data

Standard Deviation

Formula of Frequency Distribution

When multiple classes are involved, the number could be larger or smaller, but determining its distribution with unorganised data will be difficult. In such cases, the ideal number of classes can be calculated using the following frequency distribution formula:-

A different frequency distribution formula is used to determine the minimum to maximum range of values, also known as the width of the value, in order to calculate a range of data that is the minimum to maximum range of heights of students in a different class. This can be stated as follows:

h = range / number of class

The variance combines all the values of data which can be expressed as a formula:

s² = Σf (m - x?)² / n – 1

Where σ² is variance

f = frequency of the class

m = midpoint of the class

X = mean of the data

N = no of items given

It is basically the square root deviation of each of the numbers that develop the mean of the given set of data.

Also Read:

Class Mark

Frequency Polygons

Steps of Frequency Distribution Construction

Creating a frequency distribution table is simple if you follow the steps outlined below:

Step 1: Create a table with two columns, one for the title of the data you're organising and the other for frequency. [If you want to include tally marks, draw three columns.]

Step 2: Examine the data and decide whether you want to create an ungrouped frequency distribution table or a grouped frequency distribution table. If there are too many different values, the grouped frequency distribution table is usually preferable.

Step 3: Fill in the first column with the values from the data set.

Step 4: Count the number of times each item appears in the collected data. In other words, count the number of times each item appears.

Step 5: Fill in the frequency in the second column for each item.

Step 6: Finally, in the last row of the table, write the total frequency.

Points to Remember

Following are some important points:

The frequency (f) of a specific value is the number of times the value appears in the data. A variable's distribution is a pattern of frequencies, which means the set of all possible values and the frequencies associated with them.
Frequency distributions can show either the number of observations that fall into each range or the percentage of observations that fall into each range.
In the latter case, the distribution is referred to as a relative frequency distribution.
If you have a list of numbers that represent the frequency of a specific outcome in a sample, it's a useful way to organise data.
Unlike the calculation of the range, in order to produce a measured spread of data, the variance combines all the values of data in a format that can be expressed as a formula s2 = Σf (m -x?)2 / n – 1.

Sample Questions

Ques: A school held a blood donation drive. The blood groups of 30 students are listed below.
A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O
Make a frequency distribution table out of this data. (3 Marks)

Ans: The following frequency distribution table can be used to represent the above data:

Blood Groups	Number of Students
A	9
B	6
AB	3
O	12
Total	30

Ques: The weekly pocket expenses (in dollars) of a group of 25 students chosen at random are shown below.
37, 41, 39, 34, 41, 26, 46, 31, 48, 32, 44, 39, 35, 39, 37, 49, 27, 37, 33, 38, 49, 45, 44, 37, 36
Make a grouped frequency distribution table with class intervals of equal width beginning with 25 - 30, 30 - 35, and so on. Determine your weekly out-of-pocket expenses as well. (3 Marks)

Ans: The following data is represented in the table:

Weekly Expenses	Number of Students
25-30	2
30-35	4
35-40	10
40-45	4
45-50	5
Total	25

The smallest value in the given data is 26 and the largest value is 49. So, the weekly pocket expense range = 49 - 26 = $23.

Ques: Silvia and Ashley have a deck of one-to-ten number cards. They take out a number card and record the number that appears on it. They go through the process at least 12 times. They have the following values assigned to them:
5, 8, 9, 2, 3, 7, 3, 4, 5, 9, 3, 1
Create a frequency table to better organise the data. (3 Marks)

Ans: The following data is represented in the table:

Values	Frequency
1	1
2	1
3	3
4	1
5	2
6	0
7	1
8	1
9	2
10	0
Total	12

Ques: On Maple Avenue, a survey was conducted. In each of the 20 homes, people were asked how many cars were registered to their households. The following are the results:
3, 1, 4, 0, 2, 1, 5, 2, 1, 5, 4, 2, 3, 2, 0, 2, 1, 0, 3, 2.
This data should be displayed in a Frequency Distribution Table.
Determine how many vehicles each household has registered.
Divide the number of cars (x) into intervals and count the number of outcomes in each interval (frequency). (3 Marks)

Ans: The Frequency Distribution Table is created as a result.

Number of Cars	Frequency
0	3
1	4
2	6
3	3
4	2
5	2
Total	20

Ques: Find the mean deviation about the mean for the following data. (3 Marks)

Ans:

Marks obtained	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Number of students	2	3	8	14	8	3	2

The following data is represented in the table:

Marks Obtained	Number of students	Mid-points	f i x i	\|Xi-x\|	Fi\|xi-x\|
10-20	2	15	30	30	60
20-30	3	25	75	20	60
30-40	8	35	280	10	80
40-50	14	45	630	0	0
50-60	8	55	440	10	80
60-70	3	65	195	20	60
70-80	2	75	150	30	60
Total	40	-	1800	-	400

M.D. = 1/40*400=10

Ques: Calculate the mean deviation from the mean using the following data:
6, 7, 10, 12, 13, 4, 8, 12
(2 Marks)

Ans: We take it one step at a time and arrive at the following:

Mean of the given data is

X=72/8

The respective observations' deviations from the mean, 6 – 9, 7 – 9, 10 – 9, 12 – 9, 13 – 9, 4 – 9, 8 – 9, 12 – 9, or –3, –2, 1, 3, 4, –5, –1, 3

The absolute values of the deviations, are 3, 2, 1, 3, 4, 5, 1, 3

The required mean deviation about the mean is 22/8= 2.75

Ques: Calculate the mean deviation from the mean using the following data: 12, 3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0, 5
(2 Marks)

Ans: First, we must calculate the mean (x) of the given data = 200/10=20

The respective absolute values of the deviations from mean, are 2, 7, 8, 7, 6, 1, 7, 9, 10, 5, 2, 7, 8, 7, 6, 1, 7, 9, 10, 5

M.D. = 124/20 = 6.2

Ques: Determine the mean deviation from the mean for the following data:(3 Marks)

X	2	5	6	8	10	12
f	2	8	10	7	8	5

Ans: Let us make a Table of the given data and append other columns after calculations.

Xi	fi	Xi/fi	\|xi-x\|	Fi\|xi-x\|
2	2	4	5.5	11
5	8	40	2.5	20
6	10	60	1.5	15
8	7	56	0.5	3.5
10	8	80	2.5	20
12	5	60	4.5	22.5
Total	40	300	-	92

M.D. =1/40*92= 2.3

Ques: The following frequency distribution depicts the results of a test taken by 200 students.
Make a table of cumulative frequency.
Respond to the following questions.
(I) How many students got less than 50 points?
(II) How many students received a minimum of 60 points?
(4 Marks)

Marks	Number of Students
10-19	7
20-29	11
30-39	20
40-49	46
50-59	57
60-69	37
70-79	15
80-89	7

Ans: The table below depicts the cumulative frequency table.

Class Intervals	Frequency	Cumulative Frequency
10-19	7	7
20-29	11	18
30-39	20	38
40-49	46	84
50-59	57	141
60-69	37	178
70-79	15	193
80-89	7	200

(I) The number of students obtaining less than 50 marks

= the cumulative frequency of the class interval 40 - 49 = 84.

(II) The number of students obtaining at least 60 marks

= total number of students - the number of students getting less than or equal to 59

= 200 - 141

= 59

Also Read:

Probability: Concepts, Problems and Solved Examples	Sum of Probabilities
Class Mark	Probability Impossible Events
Identity Matrix	Cumulative Frequency Curve
Mean, Median and Mode	Relation Between Mean, Median, and Mode
Difference between Mean,Median and Mode	Mean of Grouped Data

Frequency Distribution Formula: Meaning and Steps

What is Frequency Distribution?

Formula of Frequency Distribution

Steps of Frequency Distribution Construction

Points to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

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Frequency Distribution Formula: Meaning and Steps

What is Frequency Distribution?

Formula of Frequency Distribution

Steps of Frequency Distribution Construction

Points to Remember

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

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SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

5.
Find: \[ \int \frac{x^2}{(x^2-1)(x^2+4)}\,dx \]

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).