Normal Distribution – Solved Problems
The normal distribution, also called the Gaussian distribution, is one of the most important probability distributions in statistics. Many real-life variables such as height, marks, and measurement errors approximately follow a normal distribution.
Problem 1: Identifying Normal Distribution
The mean height of students in a class is 160 cm with a standard deviation of 5 cm. What is the distribution used to model this data?
Solution:
When data clusters symmetrically around a mean and follows a bell-shaped curve, it is modeled using a normal distribution.
Here, mean and standard deviation are provided, so the appropriate model is a normal distribution.
Answer: The data follows a normal distribution.
Problem 2: Z-Score Calculation
The mean score in an exam is 70 and the standard deviation is 10. Find the Z-score for a student who scored 85.
Solution:
Z = (85 − 70) / 10 = 15 / 10 = 1.5
Answer: Z-score = 1.5
Problem 3: Probability Less Than a Given Value
Assume the marks in an exam are normally distributed with mean 50 and standard deviation 10. Find the probability that a randomly selected student scores less than 60.
Solution:
First calculate the Z-score:
From the standard normal distribution table:
Answer: Probability ≈ 0.841
Problem 4: Probability Between Two Values
Heights of students are normally distributed with mean 165 cm and standard deviation 5 cm. Find the probability that a student’s height lies between 160 cm and 170 cm.
Solution:
Calculate Z-scores:
From Z-table:
Answer: Probability ≈ 0.683
Problem 5: Empirical Rule Application
If the mean weight of a population is 60 kg and standard deviation is 4 kg, what percentage of people have weight between 56 kg and 64 kg?
Solution:
56 kg and 64 kg are one standard deviation away from the mean.
By the empirical rule:
Answer: Approximately 68% of people
Problem 6: Finding a Cut-Off Score
Scores in a test are normally distributed with mean 100 and standard deviation 15. Find the score above which the top 5% of students lie.
Solution:
Top 5% corresponds to Z ≈ 1.645
Answer: Cut-off score ≈ 124.7
Problem 7: Real-Life Interpretation
IQ scores are normally distributed with mean 100 and standard deviation 15. What percentage of people have an IQ greater than 130?
Solution:
Calculate Z-score:
From the Z-table:
Answer: Percentage ≈ 2.28%
Conclusion: Normal distribution plays a vital role in statistics and real-world data analysis. Understanding Z-scores, probability areas, and interpretation is essential for exams, analytics, and scientific studies.