Quantitative Aptitude · Theoretical Distributions

Normal Distribution and Its Properties for CA Foundation

Updated 1 October 2026 · Fact-checked

The normal distribution is a continuous, bell-shaped, symmetric probability distribution fully described by its mean (μ) and standard deviation (σ). Mean, median and mode are equal. About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean. To solve questions, identify μ and σ, then apply these properties.

Understand Normal Distribution and Its Properties

A normal distribution is a continuous distribution that gives the familiar bell-shaped curve. Heights, exam marks and measurement errors often follow it roughly. Most values cluster near the centre. Very high and very low values are rare.

The curve has two parameters: the mean μ and the standard deviation σ. μ decides where the centre sits. σ decides how wide or narrow the bell is. A larger σ gives a flatter, wider curve. A smaller σ gives a taller, narrower curve. If you write X ~ N(μ, σ²), you are saying X is normal with mean μ and variance σ².

The curve is symmetric about μ. So the left half is a mirror image of the right half. Because of this, mean = median = mode = μ. Skewness is zero. The curve is unimodal, which means it has one peak, at x = μ.

The total area under the curve is 1, because it represents total probability. Each half of the curve has area 0.5. For a continuous variable, the probability of one exact value is 0. Probability is an area between two points. The curve never touches the x-axis. It stretches to −∞ and +∞ and approaches the axis (it is asymptotic).

The empirical rule gives areas without tables. About 68.27% of the area lies within μ ± σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ. Questions often round these to 68%, 95% and 99.7%. The curve has points of inflection at μ − σ and μ + σ. The quartiles are at about μ ± 0.6745σ.

Key formulas to remember

Normal variable notation: X ~ N(μ, σ²); μ is the mean, σ is the standard deviation, σ² is the variance.
Probability density function: f(x) = [1 ÷ (σ√(2π))] × e^(−(x − μ)² ÷ (2σ²)), for −∞ < x < +∞; Rarely used for calculation. You may need to recognise it. The π here is 3.14159..., and e is about 2.718.
Central equality: Mean = Median = Mode = μ; True because the curve is symmetric and has a single peak.
Skewness and kurtosis: Skewness (β₁ and γ₁) = 0; β₂ = 3; β₂ = 3 means the curve is mesokurtic.
Empirical rule: P(μ − σ < X < μ + σ) ≈ 68.27%; P(μ − 2σ < X < μ + 2σ) ≈ 95.45%; P(μ − 3σ < X < μ + 3σ) ≈ 99.73%; Exact values are 68.27%, 95.45% and 99.73%. The rounded values 68%, 95% and 99.7% are usually used in the options.
Area on each side: P(X < μ) = P(X > μ) = 0.5; Total area under the curve = 1.
Quartiles: Q₁ = μ − 0.6745σ; Q₃ = μ + 0.6745σ; Quartile deviation = 0.6745σ, which is about (2/3)σ. Mean deviation is about 0.7979σ, which is about (4/5)σ.
Quartile deviation, mean deviation and SD (approximate): QD : MD : SD ≈ 10 : 12 : 15, from QD ≈ (2/3)σ and MD ≈ (4/5)σ; This ratio is only an approximation. The exact ratio 0.6745 : 0.7979 : 1 is roughly 10.1 : 12 : 15. With σ = 15, QD ≈ 10 and MD ≈ 12.
Points of inflection: x = μ − σ and x = μ + σ; The curve changes from bending downward to bending upward at these points.
Standardisation: Z = (X − μ) ÷ σ; Z has mean 0 and SD 1. Table use is covered in the Z-table topic.

How to solve Normal Distribution and Its Properties questions

Use this method for both theory-type and numerical MCQs on the normal distribution.

1Read the question and note what is given: μ, σ, or a range of values.
2Decide the type. It may be a property question (symmetry, mean, median, mode, skewness) or an area question.
3For a property question, recall the standard facts: bell shape, symmetric, unimodal, mean = median = mode, skewness 0, total area 1.
4For an area question, convert the range into distance from μ in units of σ. For example, find how many σ the limit lies above or below μ.
5If the limits are exactly 1, 2 or 3 σ from μ, use the empirical rule. If one limit is μ itself, use the half-area 0.5.
6Use symmetry to add or subtract areas. Left of μ mirrors right of μ.
7If the limits are not whole σ values, use Z = (X − μ) ÷ σ and the table values given in the question.
8Check the answer lies between 0 and 1 (or 0% and 100%), and that wider ranges give larger areas.

Quickest way: Sketch, mark μ ± σ, and read off the area

When to use it: Use this for any area or percentage question where the limits are whole multiples of σ from the mean.

Draw a quick bell curve and mark μ in the middle.
Mark μ ± σ, μ ± 2σ and μ ± 3σ on the axis.
Remember the one-sided pieces: 34% for 0 to 1σ, 13.5% for 1σ to 2σ, 2.4% for 2σ to 3σ. These are approximate and add up to about 50% per side.
Add the pieces that match the question. For example, μ to μ + 2σ is about 34% + 13.5% = 47.5%.
For a tail, subtract from 50%. For example, above μ + σ is about 50% − 34% = 16%.
Pick the option closest to your answer. If no option fits and the limits are not whole σ values, skip the question rather than guess.

Common mistakes in Normal Distribution and Its Properties

Saying the mean, median and mode of a normal distribution are different.
Students remember the general mean-median-mode relation for skewed data.
Fix: For a normal curve, symmetry and a single peak make all three equal to μ.
Treating 68%, 95% and 99.7% as areas from the mean to μ + σ, μ + 2σ, μ + 3σ.
Students forget that the rule covers both sides of the mean.
Fix: The rule is for μ ± kσ. The one-sided area from μ to μ + σ is about 34%.
Saying the curve touches the x-axis at μ ± 3σ.
Most of the area is within 3σ, so the tails look as if they end there.
Fix: The curve is asymptotic. It never touches the axis, even though almost all area lies within μ ± 3σ.
Thinking a larger σ makes the curve taller.
Students link a bigger number with a bigger picture.
Fix: Total area stays 1. A larger σ spreads values out, so the curve is flatter and wider. Smaller σ is taller and narrower.
Thinking that changing μ changes the shape of the curve.
Students mix up location and spread.
Fix: Changing μ only slides the curve left or right. Only σ changes its width and height.
Quoting P(X = a) as a positive number for a normal variable.
Students carry over the idea from discrete distributions like binomial.
Fix: For a continuous variable, probability at a single point is 0. Only intervals have positive probability.

Worked examples

Example 1

Marks in a test are normally distributed with mean 60 and standard deviation 8. Approximately what percentage of students score between 52 and 68? (A) 34% (B) 68% (C) 95% (D) 99.7%

Show the solution

μ = 60 and σ = 8.
Lower limit 52 = 60 − 8 = μ − σ.
Upper limit 68 = 60 + 8 = μ + σ.
So the range is μ ± σ.
By the empirical rule, about 68% of values lie in this range.

Answer: (B) 68%

Example 2

For a normal distribution with mean 50 and standard deviation 10, approximately what proportion of values exceeds 70? (A) 0.0228 (B) 0.0500 (C) 0.1587 (D) 0.3174

Show the solution

70 = 50 + 2 × 10, so the limit is μ + 2σ.
The exact area within μ ± 2σ is 0.9545.
Two-tail area = 1 − 0.9545 = 0.0455.
By symmetry, the upper tail is half of this: 0.0455 ÷ 2 = 0.02275 ≈ 0.0228.
Option (B) 0.0500 is the rounded two-tail area (1 − 0.95), not the one-tail value.
Option (C) 0.1587 is the tail beyond μ + σ, not beyond μ + 2σ.
Option (D) 0.3174 is the two-tail area outside μ ± σ (2 × 0.1587).

Answer: (A) 0.0228

Example 3

For a normal distribution, the quartile deviation is 6. What is the approximate value of the standard deviation? (A) 4 (B) 7.2 (C) 9 (D) 12

Show the solution

For a normal distribution, QD ≈ (2/3)σ.
So 6 = (2/3)σ.
σ = 6 × 3 ÷ 2 = 9.
Check with the exact factor: σ = 6 ÷ 0.6745 ≈ 8.9. This is very close to 9, and 9 is the nearest option.

Answer: (C) 9

Exam tips

Theory MCQs often ask which statement is true or false. Memorise: symmetric, bell-shaped, unimodal, mean = median = mode, skewness 0, total area 1.
Know the approximate relation QD ≈ (2/3)σ and MD ≈ (4/5)σ, which gives QD : MD : SD ≈ 10 : 12 : 15. It is an approximation, not an exact ratio, so pick the nearest option. It is a favourite for conversion questions.
If the question gives μ ± kσ with whole k, use the empirical rule and avoid Z-tables. This saves time.
Watch the wording: 'within' usually means both sides, while 'exceeds' or 'below' means one tail.
With 0.25 negative marking, skip an area question that needs a table value you cannot recall and no table is given.

Practice questions from Theoretical Distributions

Normal Distribution and Its Properties: frequently asked questions

What are the main properties of the normal distribution?

It is bell-shaped, symmetric about the mean, and has a single peak. Mean, median and mode are equal, skewness is 0, and total area under the curve is 1. The curve is asymptotic to the x-axis on both sides.

What is the 68-95-99.7 rule?

It says that about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. It applies to normal distributions only. Exact values are 68.27%, 95.45% and 99.73%.

What are the parameters of a normal distribution?

The two parameters are the mean μ and the standard deviation σ. The mean fixes the centre of the curve and σ fixes its spread. Once both are known, the whole curve is determined.

Is the normal distribution discrete or continuous?

It is continuous. The variable can take any real value, and probabilities are areas under the curve. The probability of any single exact value is zero.