Q.(a) If $X$ is a normal variate with mean $(\mu) = 70$ and standard deviation $(\sigma) = 5$, then find $P(X > 75)$. (Given : $P(0 < Z < 1) = 0.3413$)

Z Score: Measuring How Unusual a Value Is

Imagine you're in a class of 100 students. The test average is 60 out of 100. You scored 75. That's 15 marks above average — good. But what if the class average was 60 and the standard deviation was only 2? Then almost everyone scored between 54 and 66. Your 75 would be extraordinary — you'd be miles ahead of everyone else.

Now imagine the same average of 60, but the standard deviation is 20. Now scores range from roughly 0 to 120. Your 75 is still above average, but it's not remarkable — many students scored higher.

The Z score captures exactly this: how many standard deviations a value is from the mean. It tells you not just "how far" but "how far relative to the spread."

The Intuition in One Sentence

A Z score answers: "If the average is the center of the data, and the standard deviation is the typical distance from that center, then my value is ___ typical distances away."

• Z = 0 → exactly at the mean.
• Z = +1 → one standard deviation above the mean.
• Z = -2 → two standard deviations below the mean.

The bigger the absolute Z score, the more unusual the value.

The Precise Formula

Where:

• $x$ = the raw value (your score, height, weight, etc.)
• $\mu$ (mu) = the population mean
• $\sigma$ (sigma) = the population standard deviation

If you're working with a sample rather than a population, use the sample mean $\bar{x}$ and sample standard deviation $s$:

Worked Example

A class has test scores with mean $\mu = 70$ and standard deviation $\sigma = 8$.

Student A scores 86.

Student A is 2 standard deviations above the mean. That's quite high — in a normal distribution, only about 2.5% of students score higher.

Student B scores 62.

Student B is 1 standard deviation below the mean. That's below average but not extreme — about 16% of students score lower.

What Z Scores Are Used For …