Z-Score (Standard Score) Formula

The Formula

z = \frac{x - \mu}{\sigma}

When to use: Z-scores put everything on the same scale. A z-score of +2 means 'two standard deviations above average' - unusually high. A z-score of -1 means 'one SD below average' - somewhat low but normal.

Quick Example

Test mean=75, SD=10. Your score=95. z = \frac{95-75}{10} = +2 You're 2 SDs above average - top ~2.5% if normal.

Notation

z is the z-score, x is the raw value, \mu is the population mean, and \sigma is the population standard deviation. Z \sim N(0,1) is the standard normal variable.

What This Formula Means

A z-score tells you how many standard deviations a value is from the mean, calculated as z = \frac{x - \mu}{\sigma}. Positive z-scores are above the mean; negative z-scores are below. Z-scores allow comparison of values from different distributions.

Z-scores put everything on the same scale. A z-score of +2 means 'two standard deviations above average' - unusually high. A z-score of -1 means 'one SD below average' - somewhat low but normal.

Formal View

The z-score standardizes a value: z = \frac{x - \mu}{\sigma}. For a normal distribution, P(Z < z) is found from the standard normal table.

Worked Examples

Example 1

medium
A student scores 78 on a test where \mu = 70 and \sigma = 4. Calculate her z-score and interpret it.

Solution

  1. 1
    Step 1: z = \frac{x - \mu}{\sigma} = \frac{78 - 70}{4} = 2.
  2. 2
    Step 2: A z-score of 2 means the student scored 2 standard deviations above the mean.
  3. 3
    Step 3: By the empirical rule, only about 2.5% of students scored higher.

Answer

z = 2. The student scored 2 standard deviations above the mean.
Z-scores standardise values to a common scale, telling us how many standard deviations a value is from the mean. This allows comparison across different distributions.

Example 2

hard
Alice scores 85 in Maths (\mu = 75, \sigma = 5) and 90 in English (\mu = 80, \sigma = 10). In which subject did she perform better relative to her class?

Common Mistakes

  • Forgetting negative z-scores exist
  • Misinterpreting magnitude
  • Using with non-normal data carelessly

Why This Formula Matters

Z-scores allow comparing values from different distributions. Is being 6'2" more unusual than earning \$100k? Z-scores can answer this.

Frequently Asked Questions

What is the Z-Score (Standard Score) formula?

A z-score tells you how many standard deviations a value is from the mean, calculated as z = \frac{x - \mu}{\sigma}. Positive z-scores are above the mean; negative z-scores are below. Z-scores allow comparison of values from different distributions.

How do you use the Z-Score (Standard Score) formula?

Z-scores put everything on the same scale. A z-score of +2 means 'two standard deviations above average' - unusually high. A z-score of -1 means 'one SD below average' - somewhat low but normal.

What do the symbols mean in the Z-Score (Standard Score) formula?

z is the z-score, x is the raw value, \mu is the population mean, and \sigma is the population standard deviation. Z \sim N(0,1) is the standard normal variable.

Why is the Z-Score (Standard Score) formula important in Statistics?

Z-scores allow comparing values from different distributions. Is being 6'2" more unusual than earning \$100k? Z-scores can answer this.

What do students get wrong about Z-Score (Standard Score)?

Students interpret z-scores as percentages. A z-score of 2 does not mean 2% — you need a normal distribution table to convert it to a percentile.

What should I learn before the Z-Score (Standard Score) formula?

Before studying the Z-Score (Standard Score) formula, you should understand: standard deviation intro, mean fair share.

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