Mean, median, mode & range

An average is a single value that represents a data set. There are three, plus a measure of spread:

Mode vs median: the mode is the most frequent value, the median is the middle value once you've put the data in order — don't mix them up.

Your turn — the mean

Finding the median

The median is the middle value after ordering the data from smallest to largest. With an odd count there's one middle value; with an even count you take the mean of the middle two.

Always order first: the median of 4, 8, 6, 10, 2 is not the middle of the list as written (that would be 6 by luck) — you must reorder every time, or you'll get it wrong.

Your turn — the median

The range — measuring spread

The range tells you how spread out the data is. A small range means the values are close together; a large range means they're widely scattered.

Don't just subtract the ends of the list: find the actual largest and smallest values first — in 12, 5, 20, 8 the range is 20 − 5, not 12 − 8.

Your turn — the range

Mean from a frequency table

A frequency table records how often each value occurs. To find the mean, multiply each value by its frequency, add those up, then divide by the total frequency:

Divide by Σf, not by the number of rows: here there are 3 rows but 10 pieces of data. The mean is 23 ÷ 10 = 2.3, not 23 ÷ 3.

Your turn — mean from a table

Estimated mean from grouped data

When data is grouped into class intervals, we've lost the exact values. We estimate the mean using the midpoint of each class as a stand-in for every value in it:

Why "estimated"? We don't know the exact values in each group, so using midpoints gives an estimate, not the true mean. Always call it the estimated mean.

Your turn — estimated mean

Cumulative frequency (Higher tier)

Cumulative frequency is a running total of the frequencies. Plotting it against the upper class boundary gives an S-shaped curve you can read the median and quartiles from.

Plot at the upper boundary: a cumulative frequency point is plotted at the top end of each class (the highest value so far), never at the midpoint.

Box plots & the IQR (Higher tier)

A box plot summarises data with five numbers: minimum, lower quartile, median, upper quartile and maximum. The interquartile range (IQR) measures the spread of the middle half:

IQR beats range for spread: the range uses the extreme values, which may be outliers. The IQR ignores the outer quarters, so it's a more reliable measure of spread.

Your turn — median position

Statistic ⇄ definition

Tap a statistic name on the left, then its correct definition on the right.

Scatter graphs & correlation

A scatter graph plots two variables to reveal a relationship. Correlation describes the pattern:

• Positive correlation: as one goes up, the other goes up.
• Negative correlation: as one goes up, the other goes down.
• No correlation: no clear pattern.

Correlation ≠ cause: two things trending together doesn't prove one causes the other — there may be a hidden factor behind both.

Spot the correlation

Sampling & the population

The population is everyone or everything you're studying. A sample is a smaller group taken from it — used because surveying the whole population is often impractical.

• Random sample: every member has an equal chance — reduces bias.
• Biased sample: some groups are over- or under-represented, so conclusions are unreliable.

Bigger isn't the only thing: a large sample is good, but it must also be representative. Surveying only your friends is biased no matter how many you ask.

Why do we sample?

Which average should you use?

Each average has strengths and weaknesses. Choosing the right one is a common exam skill:

• Mean: uses every value — but is distorted by outliers (extreme values).
• Median: the middle value — not affected by outliers, good for skewed data.
• Mode: the only average that works for non-numerical data (e.g. favourite colour).

Outlier tip: if one salary in a company is huge, the mean wage looks misleadingly high. The median gives a fairer "typical" value.

Choosing an average

Positive or negative correlation?

Each scenario links two variables. Tap a scenario, then tap the box for its correlation type.

The whole Statistics strand

You've covered the Statistics strand of OCR GCSE Maths (J560) — Foundation content plus the Higher-tier cumulative frequency and box plots. Press Finish to see your score.