Leaving Cert Statistics (Higher Level): Normal Distribution & Hypothesis Testing

Statistics covers data analysis, the normal distribution, standard deviation, z-scores and hypothesis testing. It is examined on Paper 2 and links to probability.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

The normal distribution is symmetric and bell-shaped around its mean.
A z-score measures how many standard deviations a value is from the mean.
The empirical rule: about 68%, 95% and 99.7% of data lie within 1, 2 and 3 standard deviations.
Statistics is examined on Paper 2.

Statistics explained

Types of Data

Categorical Data: Data that can be divided into groups/categories - Nominal: No natural order (colors, names) - Ordinal: Natural order (rankings, ratings)

Numerical Data: Data represented by numbers - Discrete: Countable values (number of students) - Continuous: Measurable values (height, weight)

Primary and Secondary Data

Primary Data: Data you collect yourself - Surveys and questionnaires - Experiments - Observations

Secondary Data: Data collected by others - Census data - Published research - Online databases

Primary data is tailored to your needs but takes time to collect.

Sampling Methods

Simple Random Sample: Every member has equal chance of being selected

Stratified Random Sample: Divide population into groups (strata), then randomly sample from each

Cluster Sample: Divide into clusters, randomly select some clusters, survey all in selected clusters

Quota Sampling: Select specific numbers from different groups (non-random)

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Key formulas

Name	Formula	Description
Mean	$\bar{x} = \frac{\sum x}{n}$	Average of a set of values
Mean (Frequency)	$\bar{x} = \frac{\sum fx}{\sum f}$	Mean for frequency distribution
Standard Deviation	$\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}$	Measure of spread from the mean
Standard Deviation (Alt)	$\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}$	Alternative formula for standard deviation
Variance	$\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}$	Square of standard deviation
Z-Score	$z = \frac{x - \mu}{\sigma}$	Number of standard deviations from the mean

Worked examples

Worked example 1

Find the mean of: 4, 7, 2, 9, 8

Mean = (4 + 7 + 2 + 9 + 8) ÷ 5 = 30 ÷ 5 = 6.

Answer:

Worked example 2

What type of data is "student grades (A, B, C, D, F)"?

Grades have natural order (A > B > C > D > F) but are categories, making this ordinal categorical data.

Answer:

ordinal

Where students lose marks

Reading the wrong value from the standard normal tables.
Confusing population and sample standard deviation.
Setting up the hypothesis test with the wrong margin of error.

Frequently asked questions

How do I use z-scores in Leaving Cert statistics?

A z-score tells you how many standard deviations a value lies from the mean. Calculate it as (x − μ) ÷ σ, then look it up in the standard normal tables to find the probability.

What does the 68-95-99.7 rule mean for the Leaving Cert?

For a normal distribution, roughly 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This rule is used in hypothesis testing and estimation questions.

Is statistics on Paper 1 or Paper 2?

Statistics is examined on Paper 2.

Authoritative sources

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