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Agniveer (All) Statistics

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This page covers Agniveer (All) Statistics with complete concept notes, 35 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Statistics— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

STATISTICS — CORE CONCEPT Statistics is the science of collecting, organizing, and analyzing data. In NDA exams, you mainly deal with Measures of Central Tendency (Mean, Median, Mode) and Measures of Dispersion (Range, Variance, Standard Deviation).

--- CORE CONCEPT --- Measures of Central Tendency tell you the 'center' of data. Measures of Dispersion tell you how 'spread out' the data is. Both types appear regularly in NDA papers.

---

2.Key Rules

Core rules you must know cold
1

Mean is affected by extreme values (outliers). Median and Mode are not.

2

For a symmetric distribution: Mean = Median = Mode.

3

For a moderately skewed distribution: Mode = 3 Median - 2 Mean (Empirical relation).

4

Standard Deviation is always non-negative.

5

Variance = (Standard Deviation) squared.

3.Formula Block

Memorise — at least one formula appears in every paper
Arithmetic Mean (ungrouped): Mean = Sum of all values / Number of values
Mean (grouped data): Mean = Sum of (f × x) / Sum of f
where f = frequency, x = midpoint of class
If n is odd: Median = value at position (n+1)/2
If n is even: Median = average of values at positions n/2 and (n/2 + 1)
Mode = Most frequently occurring value
Variance (ungrouped): Variance = Sum of (x - Mean)^2 / n
Standard Deviation = Square root of Variance
Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100
Range = Maximum value - Minimum value

--

Median (ungrouped, sorted data):

---

4.Exam Patterns

What examiners ask — read before attempting PYQs

--- NDA asks: finding mean/median/mode from raw data or frequency tables, calculating variance and standard deviation, using the empirical formula (Mode = 3 Median - 2 Mean), and comparing two data sets using CV. Expect 2-3 questions per paper from this subtopic. --- SHORTCUT / TRICK --- Trick 1 (Step Deviation for Mean): If values are large, use: Mean = A + (Sum of f×d / Sum of f) × h, where A = assumed mean, d = (x - A)/h, h = class width. This saves huge calculation time. Trick 2 (Quick Variance): Variance = [Sum of x^2 / n] - (Mean)^2. You do NOT need to subtract the mean from each value individually.

Just find average of squares minus square of average. ---

5.Worked Example

Solve this step-by-step before moving on

-- Question: Find Mean, Variance, and Standard Deviation for: 2, 4, 6, 8, 10. Step 1 — Find Mean: Mean = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6 Step 2 — Use Variance shortcut: Sum of x^2 = 4 + 16 + 36 + 64 + 100 = 220 Mean of x^2 = 220 / 5 = 44 Variance = 44 - (6)^2 = 44 - 36 = 8 Step 3 — Standard Deviation: SD = Square root of 8 = 2√2 ≈ 2.83 Answer: Mean = 6, Variance = 8, SD = 2√2 ---

6.Exam Traps

Common mistakes students make — avoid these

--- Students often confuse Variance with Standard Deviation. Remember: Variance uses squared units. Standard Deviation is in the same unit as the data.

Also, many students forget to sort data before finding the Median — always sort first!

Key Points to Remember

  • Mean = Sum of values / Number of values; affected by extreme values (outliers).
  • Median is the middle value of sorted data; not affected by outliers.
  • Mode is the most frequent value; a data set can have more than one mode.
  • Empirical relation: Mode = 3 Median - 2 Mean (for moderately skewed data).
  • Variance = Average of squares minus square of average: Variance = (Sum x^2 / n) - Mean^2.
  • Standard Deviation = Square root of Variance; always greater than or equal to zero.
  • Coefficient of Variation = (SD / Mean) × 100; used to compare variability of two data sets.
  • For a perfectly symmetric distribution: Mean = Median = Mode.

Exam-Specific Tips

  • The empirical formula connecting Mean, Median, and Mode is: Mode = 3 Median - 2 Mean.
  • Coefficient of Variation (CV) is always expressed as a percentage: CV = (SD / Mean) × 100.
  • Variance is the square of Standard Deviation; if SD = 4, then Variance = 16.
  • If every value in a data set is increased by a constant k, the Mean increases by k but Variance and SD remain unchanged.
  • If every value in a data set is multiplied by a constant k, the Variance is multiplied by k^2 and SD is multiplied by k.
  • Range = Maximum value - Minimum value; it is the simplest measure of dispersion.
  • Standard Deviation is always less than or equal to Range of the data set.
  • For two data sets with the same mean, the one with smaller CV is more consistent.
Practice MCQs

Statistics — Practice Questions

35graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 35

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Practice 1easy

The median of the dataset {3, 7, 2, 9, 5, 8, 4} is:

Practice 2easy

The range of the dataset {15, 22, 18, 25, 20, 17, 28, 19} is:

Practice 3easy

If the standard deviation of a dataset is 4, what is the variance?

Practice 4easy

If the mean of a dataset is 20 and the standard deviation is 4, what is the coefficient of variation (in percent)?

Practice 5easy

The median of the dataset {3, 7, 5, 9, 1, 8, 4} is:

Practice 6easy

The mean of five observations is 12. If four of the observations are 8, 10, 14, and 16, what is the fifth observation?

Practice 7easy

The variance of the dataset {2, 4, 6, 8, 10} is:

Practice 8easy

Two datasets have the same mean of 50. Dataset A has a standard deviation of 5, and Dataset B has a standard deviation of 10. Which statement is correct?

Practice 9easy

The range of the dataset {3, 7, 5, 12, 8, 15, 6} is:

Practice 10easy

If the standard deviation of a dataset is 5, what is the variance?

Practice 11easy

The median of the dataset {4, 8, 2, 9, 5, 7, 6} is:

Practice 12medium

In a grouped frequency distribution, the modal class is 40–50 with frequency 20. The frequencies of the classes immediately before and after the modal class are 12 and 15 respectively. The class width is 10. Using the mode formula for grouped data, what is the mode?

Practice 13medium

A dataset has 5 observations: 10, 15, 20, 25, 30. The variance of this dataset is 50. If we remove the observation 30 and add a new observation x, the new variance becomes 40. What is the value of x?

Practice 14medium

In a dataset of 100 observations, the coefficient of variation (CV) is 20%. If the mean is 50, what is the standard deviation?

Practice 15medium

The mean and standard deviation of a dataset of 10 observations are 15 and 3 respectively. If each observation is multiplied by 2 and then 5 is added to it, what is the new standard deviation?

Practice 16medium

In a frequency distribution, the first quartile (Q₁) is 25, the median (Q₂) is 40, and the third quartile (Q₃) is 60. What is the interquartile range (IQR)?

Practice 17medium

A dataset has 25 observations with mean 50 and variance 16. If one observation of value 42 is removed, what is the new mean? (Round to 2 decimal places if necessary.)

Practice 18medium

A sample of 100 students has a mean score of 72 and a standard deviation of 8. Assuming the scores are normally distributed, approximately what percentage of students scored between 64 and 80?

Practice 19medium

The mean and standard deviation of a dataset of 10 observations are 15 and 3 respectively. If each observation is multiplied by 2 and then 5 is added to each result, what is the new standard deviation?

Practice 20medium

For a frequency distribution, the first quartile (Q₁) is 25 and the third quartile (Q₃) is 55. If a new observation of value 100 is added to the dataset, which of the following statements is necessarily true?

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60-Second Revision — Statistics

  • Formula: Mean = Sum(fx) / Sum(f) for grouped data; sort data BEFORE finding Median.
  • Shortcut: Variance = (Sum of x^2 / n) - Mean^2 — saves time, use this always.
  • Remember: Adding a constant to all values changes Mean but NOT Variance or SD.
  • Remember: Multiplying all values by k multiplies SD by k and Variance by k^2.
  • Trap: Do NOT confuse Variance and SD — Variance is SD squared, not the same thing.
  • Formula: Empirical relation — Mode = 3 Median - 2 Mean — use when one value is missing.
  • Trap: Always arrange data in ascending order before picking Median position.
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