Understanding Z-Scores and Normal Distribution

The z-score is a fundamental concept in statistics that tells you how many standard deviations a data point is from the mean of its distribution. Whether you are analysing exam results, quality control data, or scientific measurements, z-scores allow you to standardise values so they can be compared across different datasets.

The Z-Score Formula

The formula for calculating a z-score is:

Where:

• = the individual data point (the value you are examining)
• = the population mean (average)
• = the population standard deviation

A positive z-score means the value is above the mean, while a negative z-score means it is below the mean. A z-score of 0 means the value equals the mean exactly.

The Normal Distribution and the Bell Curve

The normal distribution (also called the Gaussian distribution) is a symmetric, bell-shaped curve centred on the mean. It is one of the most important probability distributions in statistics because many natural phenomena follow this pattern: heights, test scores, measurement errors, and more.

Key properties of the normal distribution:

• The mean, median, and mode are all equal and located at the centre.
• The curve is perfectly symmetric about the mean.
• The total area under the curve equals 1 (representing 100% probability).
• The tails extend infinitely in both directions but never touch the horizontal axis.

The 68-95-99.7 Rule (Empirical Rule)

One of the most useful facts about the normal distribution is the empirical rule, which states:

• 68% of data falls within of the mean (z-scores between and )
• 95% of data falls within of the mean (z-scores between and )
• 99.7% of data falls within of the mean (z-scores between and )

This means that a z-score beyond is extremely rare, occurring less than 0.3% of the time.

Worked Example 1: Exam Scores

Problem: A class has a mean exam score of 72 with a standard deviation of 8. A student scored 88. What is their z-score?

The student's score is 2 standard deviations above the mean. Using the empirical rule, we know that roughly 97.5% of students scored below this mark (since 95% falls within , leaving 2.5% above ).

Worked Example 2: Quality Control

Problem: A factory produces bolts with a mean length of 50 mm and a standard deviation of 0.3 mm. A bolt measures 49.4 mm. Is it within acceptable limits (within 2 standard deviations)?

The z-score is exactly , which sits right at the boundary of the 95% interval. This bolt is borderline and would likely be flagged for review in most quality control processes.

Worked Example 3: Comparing Across Different Tests

Problem: Sarah scored 78 on a maths test (mean 70, standard deviation 5) and 85 on an English test (mean 80, standard deviation 10). On which test did she perform relatively better?

Maths z-score:

English z-score:

Although Sarah's raw English score is higher, her maths z-score (1.6) is much greater than her English z-score (0.5). She performed relatively better in maths compared to her classmates.

How to Read a Z-Score Table

A z-score table (also called a standard normal table) gives the cumulative probability to the left of a given z-value. To use it:

1. Calculate your z-score using the formula above.
2. Find the row corresponding to the first decimal place (e.g. for , find the row for 1.6).
3. Find the column corresponding to the second decimal place (e.g. the column for 0.04).
4. The value at that intersection is the probability that a randomly chosen value falls below your z-score.

For example, gives a table value of approximately 0.9495, meaning 94.95% of the data lies below this point.

Finding the Probability Above a Z-Score

Since the total area under the curve is 1, the probability above a z-score is:

For : , or about 5.05%.

Finding the Probability Between Two Z-Scores

To find the probability between two z-scores, subtract the smaller cumulative probability from the larger one:

Tips for Working with Z-Scores

• Always check your sign. A negative z-score is not an error. It simply means the data point is below the mean.
• Z-scores work best with normally distributed data. If your data is heavily skewed, z-scores may be misleading.
• Use the sample standard deviation () instead of when working with a sample rather than the full population.
• Context matters. A z-score of 2.0 on an exam might be excellent, but the same z-score in a manufacturing process could signal a defect.

Try It Yourself

Want to calculate z-scores and explore standard deviations for your own datasets? Use our free Standard Deviation Calculator to compute the mean, variance, and standard deviation, then work out z-scores for any data point in your set. For direct z-score lookups with bell curve visualisation, try our Z-Score Calculator.