Normal Distribution

Normal Distribution is a continuous probability distribution that is symmetric about its mean, forming a bell-shaped curve. It represents how data values are distributed around the average, with most values clustering near the mean and fewer appearing as you move away. In a normal distribution, the mean, median, and mode are all equal. It is widely used in statistics, natural sciences, and social sciences to model real-world phenomena like height, test scores, and measurement errors.

1.0Define Normal Distribution

A Normal Distribution is a continuous probability distribution that is symmetric about its mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. It is also known as the Gaussian distribution.

2.0Normal Distribution Definition

A Normal Distribution is a bell-shaped probability distribution curve where the mean, median, and mode of the data are equal, and the data is symmetrically distributed around the mean.

3.0What Do You Mean by Normal Distribution?

The Normal Distribution is a statistical model that describes how data values are distributed. In many natural processes—like height, weight, test scores, etc.—the data tend to cluster around a central value with no bias left or right, forming a symmetrical bell-shaped curve.

4.0Normal Distribution Formula

The probability density function (PDF) of a normal distribution is given by:

$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$

Where:

• μ: Mean of the distribution
• σ: Standard deviation
• x: Variable
• e: Euler's number (~2.718)
• π: Pi (~3.1416)

5.0Normal Distribution Graph

• The graph is a bell-shaped curve.
• The highest point is at the mean μ.
• The area under the curve represents probability and is always equal to 1.
• The curve is symmetrical about the mean.

6.0What Are the Five Properties of a Normal Distribution?

1. Symmetry: The curve is symmetric about the mean.
2. Mean = Median = Mode: All are equal in a normal distribution.
3. Asymptotic: The tails approach the x-axis but never touch it.
4. Bell-shaped Curve: Peaks at the mean and tapers off toward both ends.
5. 68-95-99.7 Rule:
6. • 68% of data falls within 1 standard deviation
   • 95% within 2 standard deviations
   • 99.7% within 3 standard deviations

7.0Normal Distribution Examples

Example 1: Heights of People

If adult male heights are normally distributed with a mean of 175 cm and a standard deviation of 10 cm, about 68% of men have heights between 165 cm and 185 cm.

Example 2: Test Scores

In a class, scores are normally distributed with mean μ = 70 and σ = 5. Then:

• ~68% of students scored between 65 and 75.
• ~95% scored between 60 and 80.

8.0Normal Distribution Uses

Normal distribution is used in:

• Statistical inference and hypothesis testing
• Quality control and manufacturing
• Natural and social sciences
• Finance and economics
• Standardized testing (e.g., SAT, IQ scores)

9.0What Is Normal and Binomial Distribution?

10.0Solved Example on Normal Distribution 

Example 1: The scores on a math test are normally distributed with a mean of 70 and a standard deviation of 5. What percentage of students scored between 65 and 75?

Solution:

Since 65 and 75 are within 1 standard deviation from the mean:

$$\begin{gathered} \mu =70 \\ \sigma =5 \\ Range:\mu \pm \sigma =70\pm 5=[65,75] \end{gathered}$$

By the 68–95–99.7 rule, 68% of data lies within 1 standard deviation.

Answer: 68%

Example 2: Heights of adult men are normally distributed with mean 175 cm and standard deviation 10 cm. What is the probability that a randomly selected man is taller than 190 cm?

Solution:

Step 1: Find the Z-score

$Z=\frac{190-175}{10}=1.5$

Step 2: Find P(Z > 1.5) from standard normal table

P(Z > 1.5) = 1 - P(Z < 1.5) = 1 - 0.9332 = 0.0668 

Answer: 6.68%

Example 3: In a normal distribution with mean 80 and standard deviation 8, what score corresponds to the 90th percentile?

Solution:

Step 1: Find Z for 90th percentile ⇒ Z=1.28Z = 1.28 (from Z-table)

Step 2: Convert Z to raw score using:

$X=\mu +Z\sigma =80+1.28\times 8=80+10.24=90.24$

Answer: Approximately 90.24

Example 4: IQ scores are normally distributed with a mean of 100 and standard deviation of 15. What is the probability that a person has an IQ between 85 and 115?

Solution:

Step 1: Find Z-scores:

$Z_1=\frac{85-100}{15}=-1,Z_2=\frac{115-100}{15}=1$

Step 2: Find probabilities:

$P(-1<Z<1)=P(Z<1)-P(Z<-1)=0.8413-0.1587=0.6826$

Answer: 68.26%

Example 5: In a normal distribution with mean _{\mu = 60} and _{\sigma = 4}, find the probability that a randomly chosen value is less than 54.

Solution:

Step 1: $Z=\frac{54-60}{4}=-1.5$

Step 2: $P(Z<-1.5)=0.0668$

Answer: 6.68%

11.0Practice Questions on Normal Distribution

1. A set of test scores is normally distributed with mean 80 and standard deviation 6. Find the percentage of scores between 74 and 86.
2. IQ scores are normally distributed with μ = 100, σ = 15. What proportion of people have an IQ over 130?
3. A factory produces rods with length normally distributed, mean 20 cm and σ = 0.5. What's the probability a rod is longer than 21 cm?