The normal distribution · Grade 11 · 4 units
In The normal distribution (Grade 11 · 4 units) you practise 9 skills: The z score · The shape of the bell curve · Reading the normal table · The proportion between two values · From a proportion to a value · Finding the mean and the standard deviation · Comparing two tests · How many marks in a range · The normal distribution and the rules of probability. In a normal distribution the values gather around the mean, and its graph is a symmetric bell curve: its peak is above the mean, and the larger the standard deviation, the wider and lower the bell. The whole area under the curve is one, and the area above an interval is the proportion of values in it.
The z score of a value says how many standard deviations it lies from the mean: subtract the mean and divide by the standard deviation. A positive z score is above the mean, a negative one below it. About 68% of the values lie within one standard deviation of the mean, and about 95% within two. The table gives, for each z score, the area to its left, the proportion of values below it. The area to the right is one minus the area to the left, the area between two values is the difference of their areas, and by symmetry the share below a negative z score equals the share above the positive one. Going the other way, find in the table the area nearest to the given proportion, read off the z score, and work out the value: the mean plus the z score times the standard deviation. In the same way you can find a missing mean or standard deviation, compare marks in different tests, and count how many values lie in a range.
Tip: The table gives the area to the left of the z score.
What you learn
The z score
For example: Marks in a test have mean 46 and standard deviation 20 . What is the z score of a mark of 6 ?
Answer and solution
The answer: −2
Put it in: z = (6 − 46) ÷ 20 , that is z = −2 .
The shape of the bell curve
For example: Some values are normally distributed. Which statement is true?
- The standard deviation of the z scores is 0.
- The mean of the z scores is 1.
- The peak of the curve is above the standard deviation.
- About half of the values lie above the mean.
Answer and solution
The answer: About half of the values lie above the mean.
About half of the values lie above the mean. The curve is symmetric about the mean, so half the area under it lies to the right of the mean.
Reading the normal table
For example: The variable Z is standard normal, that is with mean 0 and standard deviation 1 . What is P(Z < 0.39) as a percentage, to 1 decimal place?
| z | P(Z < z) |
|---|---|
| 0.39 | 0.6517 |
Answer and solution
The answer: 65.2%
From the table: P(Z < 0.39) = 0.6517 , that is 65.2% .
The proportion between two values
For example: Marks in a test are normally distributed with mean 46 and standard deviation 25 . What is the probability that a mark chosen at random lies between 35 and 57 , as a percentage, to 1 decimal place?
| z | P(Z < z) |
|---|---|
| 0.44 | 0.6700 |
Answer and solution
The answer: 34%
The ends have z scores −0.44 and 0.44 . By symmetry the area between them is 2 × 0.6700 − 1 = 0.3400 , that is 34% .
From a proportion to a value
For example: The variable Z is standard normal. Above which value of z do the top 5% lie? Answer from the nearest row of the table.
| z | P(Z < z) |
|---|---|
| 1.63 | 0.9484 |
| 1.64 | 0.9495 |
| 1.65 | 0.9505 |
Answer and solution
The answer: 1.64
The proportion below the value is 0.9500 . The nearest row is z = 1.64 , with 0.9495 . The row z = 1.65 is just as near, and the smaller of the two is taken.
Finding the mean and the standard deviation
For example: Marks in a test are normally distributed with standard deviation 5 . It is known that 95% of the marks are below 48 . What is the mean? Answer to 1 decimal place.
| z | P(Z < z) |
|---|---|
| 1.63 | 0.9484 |
| 1.64 | 0.9495 |
| 1.65 | 0.9505 |
Answer and solution
The answer: 39.8
The proportion below the value is 0.9500 . The nearest row is z = 1.64 , with 0.9495 . The row z = 1.65 is just as near, and the smaller of the two is taken. Put it in x = μ + z × σ : 48 = μ + 1.64 × 5 . So μ = 39.8 .
Comparing two tests
For example: Marks in the first test are normally distributed with mean 54 and standard deviation 20 . Marks in the second test are normally distributed with mean 55 and standard deviation 10 . One mark in the first test is 85 , and one mark in the second test is 64 . Which is better compared with the other marks in its own test?
- The mark in the second test
- They are equally good
- The mark in the first test
Answer and solution
The answer: The mark in the first test
The z scores: z = (85 − 54) ÷ 20 = 1.55 in the first test, and z = (64 − 55) ÷ 10 = 0.90 in the second. The z score 1.55 is larger than 0.90 , so the mark in the first test is relatively better.
How many marks in a range
For example: A test was sat by 1,000 pupils, and their marks are normally distributed with mean 45 and standard deviation 20 . How many of them scored below 9 ?
| z | P(Z < z) |
|---|---|
| 1.80 | 0.9641 |
Answer and solution
The answer: 36
The z score: z = (9 − 45) ÷ 20 = −1.80 . From the table: P(Z < −1.80) = 1 − 0.9641 = 0.0359 . The number: 1000 × 0.0359 = 35.9 , that is about 36 .
The normal distribution and the rules of probability
For example: Marks in the first test are normally distributed with mean 63 and standard deviation 12 , and marks in the second test with mean 65 and standard deviation 15 . One mark is chosen at random from each test, independently. What is the probability that both marks are above 78 , as a percentage, to 1 decimal place?
| z | P(Z < z) |
|---|---|
| 0.87 | 0.8078 |
| 1.25 | 0.8944 |
Answer and solution
The answer: 2%
In the first test: z = (78 − 63) ÷ 12 = 1.25 . From the table P(Z < 1.25) = 0.8944 , so P(X > 78) = 1 − 0.8944 = 0.1056 . In the second test: z = (78 − 65) ÷ 15 = 0.87 . From the table P(Z < 0.87) = 0.8078 , so P(X > 78) = 1 − 0.8078 = 0.1922 . The marks are independent, so multiply: 0.1056 × 0.1922 ≈ 0.02030 , that is 2% .
Not on the site yet
Parts of this topic that are not built yet: The whole table (the site shows the rows each question needs) · Where the formula of the density comes from · Distributions that are not symmetric.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).