A discrete frequency distribution is a table that shows all the possible values a discrete variable can take, along with how many times each value occurs. In simple terms, it lists every unique outcome, category, or variable value, and the number of times it appears in the data.

A discrete variable is one that can only take specific, separate values — not values in between. The data collected for such variables is called discrete data.

In frequency-type data, we are mainly interested in knowing how often each value or category occurs. The table groups this data into mutually exclusive categories, meaning each value belongs to only one group and does not overlap with others.

Example: Number of students in a classroom or the number of children in each family of a village.

In this math article, we will study Discrete Frequency Distribution.

Discrete Frequency Distribution Definition

In statistics, we deal with different kinds of data in huge amounts. Here we use different methods to draw conclusions from the given data. We require the data to be arranged in such a way that one can perform suitable statistical experiments. Thus we group into two ways, namely, discrete and continuous frequency distribution. These groupings help us in finding mean, median and mode. Now, let us define discrete frequency distribution.

Discrete Frequency Distribution is defined as the grouping of a discrete data provided with respect to their frequencies. Here we calculate the frequency of the given discrete data and draw conclusions accordingly. The values of the variable are decided individually in a discrete frequency distribution. The frequency of a variable or observation is determined by the number of its occurrence. Discrete Frequency Distribution is also known as Ungrouped Frequency Distribution.

Discrete Frequency Distribution Formula 

When we have a discrete frequency distribution, we often need to calculate measures like the mean, variance, or standard deviation. These formulas make it easier to do that step by step.

1. Mean (Average) Formula
   Formula:

x̄ = Σ(x × f) / Σf

Where:

• x = each value in the data
• f = frequency (how many times the value appears)
• Σ(x·f) = total of each value multiplied by its frequency
• Σf = total number of observations

In simple words: Multiply each value by how often it appears, add them all together, and divide by the total frequency. That gives you the average.

2. Variance Formula
   Formula:

σ² = Σ[f × (x − x̄)²] / Σf

Steps:

1. First, find the mean (x̄).
2. Subtract the mean from each value (x − x̄).
3. Square the result.
4. Multiply each squared value by its frequency (f).
5. Add them all up and divide by the total frequency.

3. Standard Deviation Formula

Formula:

σ = √[ Σ(f × (x − x̄)²) / Σf ]

Where:

• σ = standard deviation
• x = each value
• x̄ = mean
• f = frequency
• Σ = sum of all values in the expression

Discrete Frequency Distribution Table

A frequency distribution table is a graph that illustrates how often each item in a data set occurs. Let a given data consist of n different observations such as \(\begin{array}{l}x_1,\ x_2,\ …,\ x_n\end{array} \), each with a frequency of \( f_1,\ f_2,\ …,\ f_n \). Then, as shown below, this data can be represented in tabular form containing variates and corresponding frequencies, resulting in a Discrete Frequency Distribution Table.

The occurrences of data values such as observation \( x_1 \) occur \( f_1 \) times, \( x_2 \) occur \( f_2 \) times, and so on are described in the above table.

How to construct a Discrete Frequency Distribution?

We use the process of tally chart for the construction of a discrete frequency distribution. A tally chart is a simple way of counting and recording the frequency of any given data. The following steps are to be followed.

Step 1: Draw a table with three columns.

Step 2: Write each item given in the first column only once. There should be no repetition here.

Step 3: Then we go through the given discrete data one by one and put one standing line for each repetition in the second column for each individual item. This goes till we have four standing lines and then we slash these four lines to form a group of five.

Step 4: In this way we count the number of occurrences of each individual item and then write the frequency in the third column.

For example, we have been given the following discrete data and we need to construct the discrete frequency distribution.

bus, bus, bus, walk, bus, walk, bike, bus, walk, bus, car, car, walk, walk, walk, walk, bus, bus, bike, bus, car, walk, bus, walk, bus, bike, walk, bike, bike, car, walk, walk, car, walk, bike, bus, walk, walk, car, car.

Here we will form the table and find the frequency as shown below.

 Thus we get the frequency though the tally chart. In general we use the tally chart method just for understanding and we directly write the frequency of each variate by counting verbally This saves time and there is then no need for the second column.

Mean of Discrete Frequency Distribution

Mean is defined as the average of the given numbers and is calculated by dividing the sum of given variates by the total frequency.

In discrete frequency distribution the mean may be computed by any one of the following methods.

• Direct Method
• Short-cut Method
• Step deviation Method

Here we will see the Direct Method only, which is the easiest and simplest way to calculate the mean.

Direct Method

In order to find the arithmetic mean of a discrete frequency distribution, we may use the following algorithm.

Step1: Prepare the frequency table in such a way that its first column consists of the values of the variable and the second column consists of the corresponding frequencies.

Step 2: Multiply the frequency of each row with the corresponding values of variable to obtain third column containing \( f_ix_i \)

Step 3: Find the sum of all entries in third column to obtain the value of \( \sum_{i=1}^nf_ix_i \)

Step 4: Find the sum of all the frequencies in second column to obtain the value of \( \sum_{i=1}^nf_i=N \)

Step 5: Finally use the formula \( \overline{X}=\frac{\sum_{i=1}^nf_ix_i}{N} \).

Here, \( \overline{X} \) is the required mean, N is the sum of all the frequencies given, \( x_i \) is the individual variate and \( f_i \) is the frequency of the individual variate.

For example, we want to find the mean of the following frequency distribution.

So we design the table accordingly.

\( \therefore\overline{X}=\frac{\Sigma f_ix_i}{N}=\frac{1478}{107}=13.81 \), which is the required mean.

Mode for Discrete Frequency Distribution

Mode of a distribution is that value of the variable for which the frequency is maximum.

Thus for a given discrete frequency distribution, we find the variate which has the highest frequency. That gives us the required mode.

For example, for a discrete frequency distribution given below we will find the mode.

So from the above table we see that the variate 12 has maximum frequency 70 among the rest of them. Thus the required mode for this discrete frequency distribution is 12.

Median for Discrete Frequency Distribution

Median of a distribution is the value of the variable which divides it into two equal parts i.e., it is the middle value of a distribution.

In case of a discrete frequency distribution \( \frac{x_i}{f_i} \), \( i=1,\ 2,\ 3,\ …,\ n \), we calculate the median by using the following algorithm. Here \( x_i \) is the individual variates given and \( f_i \) is the corresponding frequency of the variate.

Step 1: Find the cumulative frequency (c.f.) i.e., the cumulative addition of the frequencies one by one.

Step 2: Find \( \frac{N}{2} \), where N is the total sum of the given frequencies.

Step 3: Check the cumulative frequency which is just greater than \( \frac{N}{2} \) and determine the corresponding value of the variable.

The value thus obtained is the required median.

For example, below we have a discrete frequency distribution for which we will find the median.

As we have the sum of all the given frequencies as N=147.

Thus, \(\frac{N}{2}\)=\(\frac{147}{2}=73.5\)

Now the cumulative frequency just greater than 73.5 is 140 and the corresponding value of x is 12.

Thus the required median is 12.

Difference between Continuous and Discrete Frequency Distribution

The difference between Continuous and Discrete Frequency Distribution are listed below.

Uses of Discrete Frequency Distribution

Discrete Frequency Distribution is a part of statistics which plays an important role in real life. The uses of Discrete Frequency Distribution are listed below.

• It is a method of displaying disorganized data, such as election results, people’s income for a specific location, product sales during a specific time period, and so on.
• In finance, analysts and investors collect data about companies, industries, and market for price and volume. They use different types of frequency distributions to organize the data.
• This distribution also helps in organizing random data like daily rainfall or daily temperature and draws out conclusions accordingly.

Properties of Discrete Frequency Distribution 

A Discrete Frequency Distribution is a way of organizing data where the values are separate and countable. Here are its main properties explained in easy language:

1. Data is in Countable Form

• The values we use are separate numbers that can be counted.
• Example: Number of children in a family can be 0, 1, 2, 3, etc., but not 2.5.

2. Data is Arranged in a Table

• The values (called variables) are listed in one column, and the number of times each value appears (frequency) is listed in another column.

3. No Overlapping of Values

• Each value in the table is unique and does not repeat in another row.

4. Frequencies Display the Count

• The frequency indicates the number of times a specific value appears in the data.

5. The sum of all counts is the total frequency.

• The total number of observations in the data set can be obtained by adding up all of the frequencies.

6. Values are Fixed and Distinct

• The values do not merge into each other; they remain separate.

Example:

Here, "Number of Books" is the discrete variable, and "Number of Students" is the frequency.

Discrete Frequency Distribution Solved Examples

Example 1: Find mean, median and mode for the following dataset.

Solution:

So for finding the mean we need the product of the variate and corresponding frequency and for the median we need cumulative frequency.

\( \therefore\overline{X}=\frac{\Sigma f_ix_i}{N}=\frac{1956}{77}=25.404 \), which is the required mean.

As we have \( N=\Sigma f_i=77 \)

\( \therefore\frac{N}{2}=\frac{77}{2}=38.5 \)

Here 55 is just greater than 38.5

Thus the median is 28.

Now we see that for the variate 28, the frequency is maximum i.e., 24. Therefore the mode for this given distribution is 28.

Example 2: Find the mode and median for the following distribution.

 Solution:

First we need to calculate the frequency distribution according to the tally marks given.

So we have \( N=\Sigma f_i=52 \)

\( \therefore\frac{N}{2}=\frac{52}{2}=26 \)

So from the table we see 36 is just greater than 26. Hence the required median is 4.

Also as variate 4 has the maximum frequency 19, hence the required mode is also 4.

JEE Main Practice Questions: Discrete Frequency Distribution

Q1. Consider the following discrete frequency distribution: | x_i | 2 | 4 | 6 | 8 | 10 | |---|---|---|---|---|----| | f_i | 3 | 5 | 10 | 5 | 2 |

The mean (\bar{x}) of the distribution is:

A) 5

B) 6

C) 7

D) 6.5

Answer: B Explanation: Mean \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \sum f_i x_i = (2 \times 3) + (4 \times 5) + (6 \times 10) + (8 \times 5) + (10 \times 2) = 6 + 20 + 60 + 40 + 20 = 146 \sum f_i = 3 + 5 + 10 + 5 + 2 = 25 \bar{x} = \frac{146}{25} = 5.84 Correction: Let's re-evaluate options or adjust values to get an integer for easier checking. Let's adjust x_i=6, f_i=10 -> 60. x_i=4, f_i=5 -> 20. Let's use specific simple values: Mean \bar{x} = \frac{\sum f_i x_i}{N}. The distribution is symmetric around 6. Calculated value: 5.84. Let's check symmetry. Symmetry: 3, 5, 10, 5, 2. No, 3 vs 2 at ends. Let's assume the question intends for the student to calculate exactly. Wait, let's fix the question data to match Option B (6) perfectly for simplicity. If f for 2 is 2 instead of 3: Sum f = 24. Sum fx = 4+20+60+40+20 = 144. 144/24 = 6. Revised Data for Q1: | x_i | 2 | 4 | 6 | 8 | 10 | |---|---|---|---|---|----| | f_i | 2 | 5 | 10 | 5 | 2 | Now, \bar{x} = 6.

Q2. For a discrete frequency distribution, the cumulative frequency is useful for determining the:

A) Mean

B) Mode

C) Median

D) Range

Answer: C Explanation: The median corresponds to the value at the cumulative frequency of N/2. Cumulative frequencies are specifically constructed to locate the median position in ordered data.

Q3. If the mean of the following distribution is 2.6, find the value of the missing frequency p. | x | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | f | 4 | 5 | p | 1 | 2 |

A) 3

B) 8

C) 6

D) 2

Answer: B Explanation: \sum f = 4 + 5 + p + 1 + 2 = 12 + p \sum fx = (1 \times 4) + (2 \times 5) + (3 \times p) + (4 \times 1) + (5 \times 2) = 4 + 10 + 3p + 4 + 10 = 28 + 3p Mean = \frac{28 + 3p}{12 + p} = 2.6 28 + 3p = 2.6(12 + p) 28 + 3p = 31.2 + 2.6p 0.4p = 3.2 p = 8

Q4. The mode of the following frequency distribution is: | x_i | 10 | 20 | 30 | 40 | 50 | |---|----|----|----|----|----| | f_i | 8 | 12 | 20 | 15 | 7 |

A) 20

B) 30

C) 40

D) 25

Answer: B Explanation: In a discrete frequency distribution, the mode is the variate (x_i) corresponding to the maximum frequency. Here, the maximum frequency is 20, which corresponds to x_i = 30.

Q5. If every observation in a discrete frequency distribution is increased by 5, then the variance of the new distribution:

A) Increases by 5

B) Increases by 25

C) Remains the same

D) Becomes 5 times the original variance

Answer: C Explanation: Variance is independent of the change of origin. If we add a constant to every observation, the spread of the data (dispersion) does not change, only the location shifts. Therefore, variance remains unchanged.

Q6. Calculate the median of the data: | x | 5 | 10 | 15 | 20 | 25 | |---|---|---|---|---|---| | f | 2 | 3 | 8 | 4 | 3 |

A) 10

B) 15

C) 12.5

D) 20

Answer: B Explanation: Construct Cumulative Frequency (CF) table: x=5, f=2, CF=2 x=10, f=3, CF=5 x=15, f=8, CF=13 x=20, f=4, CF=17 x=25, f=3, CF=20 N = 20. \frac{N}{2} = 10. The cumulative frequency just greater than or equal to 10 is 13, which corresponds to x = 15.

Q7. The sum of the squares of deviations of variates from their mean is:

A) Zero

B) Maximum

C) Minimum

D) Undefined

Answer: C Explanation: A property of the mean is that the sum of the squared deviations of the observations from their arithmetic mean is minimum (Least Squares Property). Note: The sum of deviations (not squared) is zero.

Q8. Which of the following is NOT a measure of central tendency?

A) Mean

B) Median

C) Mode

D) Standard Deviation

Answer: D Explanation: Mean, Median, and Mode are measures of central tendency (location). Standard Deviation is a measure of dispersion (spread).

Q9. Calculate the Mean Deviation about the Mean for the data: x_i: 3, 9; corresponding f_i: 2, 2. Mean = 6.

A) 0

B) 3

C) 6

D) 1.5

Answer: B Explanation: Data points are 3, 3, 9, 9. Mean \bar{x} = \frac{3(2) + 9(2)}{4} = \frac{6+18}{4} = 6. Deviations |x_i - \bar{x}|: |3-6|=3, |9-6|=3. \sum f_i |x_i - \bar{x}| = 2(3) + 2(3) = 6 + 6 = 12. Mean Deviation = \frac{1}{N} \sum f_i |x_i - \bar{x}| = \frac{12}{4} = 3.

Q10. In a discrete frequency distribution, if the variance is 16 and the mean is 20, the coefficient of variation (C.V.) is:

A) 20%

B) 80%

C) 4%

D) 25%

Answer: A Explanation: Variance (\sigma^2) = 16 \Rightarrow Standard Deviation (\sigma) = \sqrt{16} = 4. Coefficient of Variation (C.V.) = \frac{\sigma}{\bar{x}} \times 100 C.V. = \frac{4}{20} \times 100 = \frac{1}{5} \times 100 = 20\%.

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