NIMCET Maths Statistics: Mean, Median, Mode, Variance & Standard Deviation

Statistics is an important part of the Probability and Statistics section of NIMCET Mathematics. It helps students understand numerical data through measures such as mean, median, mode, variance and standard deviation. These concepts are useful for solving questions involving frequency distributions, central values and the spread of observations.

NIMCET syllabus includes arithmetic mean, geometric mean, harmonic mean, median, mode, mean deviation, standard deviation, variance, moments and frequency distributions. Students preparing for NIMCET 2027 can use the NIMCET syllabus based on the 2026 examination as a preparation reference until the official 2027 syllabus is released.

What is Statistics in NIMCET Maths?

Statistics deals with collecting, organising, analysing and presenting numerical data. It helps summarise large sets of observations and identify patterns in the information.

Statistical measures are broadly divided into two categories:

1. Measures of central tendency: These identify the central or representative value of a data set.

2. Measures of dispersion: These explain how widely the observations are spread around a central value.

For NIMCET, students should understand the formulas and applications of arithmetic mean, median, mode, mean deviation, variance and standard deviation. Frequency distributions and calculations involving combined data also require practice.

Types of Data and Frequency Distribution

Before applying a statistics formula, identify the type of data given in the question. The calculation method changes depending on whether the observations are listed individually, repeated with frequencies or arranged in class intervals.

For example, if the value 2 occurs three times and the value 3 occurs twice, the total number of observations is:

N = Σfᵢ = 3 + 2 = 5

Here, N represents the total frequency and fᵢ represents the frequency of an observation.

For grouped data, calculations may involve class intervals and class marks. A class mark is the midpoint of a class interval:

Class mark = (Lower class limit + Upper class limit) / 2

Students should also check whether class intervals are continuous before applying grouped median and mode formulas.

Arithmetic Mean Formula

The arithmetic mean is the sum of all observations divided by the total number of observations. It is commonly called the average and is represented by x̄.

• Mean for Ungrouped Data

For observations x₁, x₂, x₃, ..., xₙ, the arithmetic mean is:

Mean (x̄) = (x₁ + x₂ + x₃ + ... + xₙ) / n

Here, n represents the total number of observations.

Example: Find the mean of 2, 4 and 6.

Mean = (2 + 4 + 6) / 3

Mean = 12 / 3 = 4

• Mean for Frequency Distribution

When an observation xᵢ occurs fᵢ times, multiply each observation by its frequency. Divide the sum of these products by the total frequency.

Mean (x̄) = Σfᵢxᵢ / Σfᵢ

Example: If 2 occurs three times and 3 occurs twice:

Mean = [2(3) + 3(2)] / (3 + 2)

Mean = (6 + 6) / 5

Mean = 12 / 5 = 2.4

• Assumed Mean Method

The assumed mean method simplifies calculations when the observations are large. Select a value close to the centre of the data and represent it by a.

Follow these steps:

1. Select an assumed mean (a).

2. Calculate the deviation of each observation from the assumed mean: dᵢ = xᵢ − a.

3. Multiply each deviation by its corresponding frequency.

4. Calculate the sum of the frequency-weighted deviations.

5. Apply the formula.

Mean (x̄) = a + (Σfᵢdᵢ / Σfᵢ)

For the step-deviation method, use:

uᵢ = (xᵢ − a) / h

Mean (x̄) = a + h × (Σfᵢuᵢ / Σfᵢ)

Here:

• a = Assumed mean

• dᵢ = Deviation from the assumed mean

• uᵢ = Step deviation

• h = Common factor or class width, as applicable

• Combined Mean

Combined mean is used when two or more groups are considered together. It is not always correct to calculate it by simply adding the individual means and dividing by the number of groups because the group sizes may differ.

For two groups with sizes n₁ and n₂ and means x̄₁ and x̄₂:

Combined Mean (x̄c) = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

Multiply each group mean by its group size, add the products and divide the result by the total number of observations.

The combined mean lies between the two group means and is closer to the mean of the larger group.

Students can also revise related Mathematics topics through NIMCET Set Theory and Relations and NIMCET Relations and Functions.

Mean, Median and Mode

Mean, median and mode are the three main measures of central tendency. They represent the centre of a data set in different ways.

• Mean: The average of all observations.

• Median: The middle value after arranging observations in ascending or descending order.

• Mode: The observation that occurs most frequently.

Median for Ungrouped Data

Arrange all observations in ascending order before calculating the median.

When the number of observations is odd:

Median = [(n + 1) / 2]th observation

When the number of observations is even:

Median = Average of the (n/2)th and (n/2 + 1)th observations

Example 1: Odd number of observations

Data: 2, 3, 4, 7, 8, 9, 10

Number of observations = 7

Median position = (7 + 1) / 2 = 4th observation

Median = 7

Example 2: Even number of observations

Data: 1, 2, 3, 7, 8, 11, 12, 15, 20, 22

Number of observations = 10

Median = (5th observation + 6th observation) / 2

Median = (8 + 11) / 2 = 9.5

Median for Grouped Data

For grouped frequency distributions, prepare a cumulative frequency column and identify the median class.

The median class is the class interval whose cumulative frequency first becomes equal to or greater than N/2.

Median = l + [(N/2 − F) / f] × h

Where:

The class intervals must be continuous before applying this formula.

Mode

Mode is the observation that occurs most frequently in a data set. For ungrouped data, the value with the highest frequency is the mode.

For grouped data, the class interval with the highest frequency is called the modal class.

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Here:

• l = Lower boundary of the modal class

• f₁ = Frequency of the modal class

• f₀ = Frequency of the class preceding the modal class

• f₂ = Frequency of the class succeeding the modal class

• h = Class width

For a moderately skewed distribution, the empirical relationship is:

Mode ≈ 3 × Median − 2 × Mean

This relationship can help estimate one measure when the other two are known. It is an approximate relationship, not a universal formula.

Discontinuous Class Intervals

Grouped median and mode formulas require continuous class intervals. If the given intervals are discontinuous, convert them into continuous class boundaries before applying the formulas.

In this example, 0.5 is subtracted from each lower limit and added to each upper limit. The adjustment depends on the gap between the class limits, so the same correction should not be applied to every question.

Mean Deviation

Mean deviation measures the average absolute distance of observations from a selected central value. This central value may be the mean, median or mode.

For ungrouped data, mean deviation about a central value a is:

MD(a) = Σ|xᵢ − a| / n

For a frequency distribution:

MD(a) = Σfᵢ|xᵢ − a| / Σfᵢ

Always use absolute values when calculating mean deviation. This means negative differences are also treated as positive values.

Example: Find the mean deviation about the mean for the following data:

1, 2, 3, 5, 6, 7, 8, 8, 9, 1

Mean = 50 / 10 = 5

The sum of absolute deviations from the mean is:

|1 − 5| + |2 − 5| + |3 − 5| + |5 − 5| + |6 − 5| + |7 − 5| + |8 − 5| + |8 − 5| + |9 − 5| + |1 − 5| = 26

Mean Deviation = 26 / 10 = 2.6

For 100 consecutive positive integers (1 to 100), the mean is 50.5, and the mean deviation about the mean is 25.

Variance and Standard Deviation

Variance and standard deviation measure the spread of observations around the arithmetic mean. Variance uses squared deviations, whereas standard deviation is the positive square root of variance.

Variance Formula

For ungrouped data:

Variance (σ²) = Σ(xᵢ − x̄)² / n

For a frequency distribution:

Variance (σ²) = Σfᵢ(xᵢ − x̄)² / Σfᵢ

The shortcut formula for ungrouped data is:

Variance (σ²) = (Σxᵢ² / n) − x̄²

For frequency data:

Variance (σ²) = (Σfᵢxᵢ² / Σfᵢ) − [Σfᵢxᵢ / Σfᵢ]²

These shortcut formulas are useful when the sum of observations and the sum of their squares are already available.

Standard Deviation Formula

Standard deviation is represented by σ and is calculated by taking the positive square root of variance.

Standard Deviation (σ) = √(σ²)

Some useful variance and standard deviation results for number sequences are:

1. First n natural numbers (1, 2, 3, ..., n)

Variance = (n² − 1) / 12

Standard Deviation = √[(n² − 1) / 12]

2. First n odd numbers (1, 3, 5, ..., 2n − 1)

Variance = (n² − 1) / 3

Standard Deviation = √[(n² − 1) / 3]

3. First n even numbers (2, 4, 6, ..., 2n)

Variance = (n² − 1) / 3

Standard Deviation = √[(n² − 1) / 3]

Correction of Wrong Observations

When an observation is recorded incorrectly, both the total sum and the sum of squares must be corrected before calculating the new mean and variance.

Suppose the correct observation is a, but it was recorded as b.

The corrected sum is:

S(new) = S(old) − b + a

The corrected sum of squares is:

Q(new) = Q(old) − b² + a²

Example: The mean and variance of 10 observations are 15 and 16, respectively. One observation was recorded as 25 instead of 15.

Step 1: Calculate the original sum.

Original sum = 10 × 15 = 150

Step 2: Correct the sum.

Corrected sum = 150 − 25 + 15 = 140

New Mean = 140 / 10 = 14

Step 3: Calculate the original sum of squares.

Variance = (Sum of squares / n) − (Mean)²

16 = (Q / 10) − 15²

Q / 10 = 16 + 225 = 241

Original sum of squares = 2,410

Step 4: Correct the sum of squares.

Corrected sum of squares = 2,410 − 25² + 15²

= 2,410 − 625 + 225

= 2,010

Step 5: Calculate the new variance.

New Variance = (2,010 / 10) − 14²

= 201 − 196

= 5

New Standard Deviation = √5

Combined Variance

Combined variance is calculated when two groups with different means and variances are merged. It depends on the size, mean and variance of each group.

For two groups, the combined mean is:

x̄c = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

The combined variance is:

σc² = [n₁σ₁² + n₂σ₂²] / (n₁ + n₂) + [n₁n₂ / (n₁ + n₂)²] × (x̄₁ − x̄₂)²

Here:

• n₁ and n₂ = Sizes of the two groups

• x̄₁ and x̄₂ = Means of the two groups

• σ₁² and σ₂² = Variances of the two groups

• σc² = Combined variance

Combined Variance When Both Groups Have Equal Sizes

When n₁ = n₂, the formula becomes:

σc² = (σ₁² + σ₂²) / 2 + (x̄₁ − x̄₂)² / 4

Combined Variance When Both Groups Have the Same Mean

When x̄₁ = x̄₂, the formula becomes:

σc² = (n₁σ₁² + n₂σ₂²) / (n₁ + n₂)

Example: Two groups have 15 observations each. Their means are 12 and 14, and their variances are 14 and σ², respectively. If their combined variance is 13, find the unknown variance.

Since both groups have equal sizes, use:

Combined Variance = (σ₁² + σ₂²) / 2 + (x̄₁ − x̄₂)² / 4

13 = (14 + σ²) / 2 + (12 − 14)² / 4

13 = (14 + σ²) / 2 + 1

12 = (14 + σ²) / 2

24 = 14 + σ²

σ² = 10

Therefore, the unknown variance is 10.

Statistics Formulas for NIMCET

The following formulas are useful for quick revision before practising questions or attempting a mock test.

NIMCET Maths Preparation for Statistics

Mathematics carried the highest weightage in the confirmed NIMCET 2026 examination pattern, with 50 questions carrying 600 marks. The examination also included negative marking, making accuracy an important part of preparation.

While practising statistics questions, students should focus on understanding the conditions for applying each formula rather than memorising expressions without understanding their use.

Common Mistakes to Avoid

Avoid these common mistakes while solving NIMCET Maths Statistics questions to improve calculation accuracy and apply the correct formulas.

• Applying a formula without identifying whether the data is ungrouped or grouped.

• Not arranging observations before calculating the median.

• Selecting the wrong median class or modal class.

• Forgetting to use absolute values in mean deviation.

• Not multiplying deviations by their corresponding frequencies.

• Using incorrect class boundaries in grouped median and mode questions.

• Forgetting to correct the sum of squares when an observation is recorded incorrectly.

• Confusing variance with standard deviation.

• Applying the wrong combined variance formula when group sizes or means are equal.

The official NIMCET 2027 notification and final syllabus are awaited. Until they are released, students can use the confirmed 2026 syllabus as a preparation reference and follow the NIMCET 2027 Mathematics preparation strategy. For additional practice, students can solve NIMCET previous year question papers and attempt mock tests to improve calculation speed and accuracy.

Combined variance considers the individual group variances, group sizes and the difference between their means.

Mean, median and mode represent the centre of data, while mean deviation, variance and standard deviation explain its spread. Regular practice of these statistics formulas can help students solve NIMCET Mathematics questions with better speed and accuracy.

For complete NIMCET preparation, students can explore PW's NIMCET online batches, which provide live and recorded classes, practice questions, mock tests and revision support for Mathematics and other examination sections.