Describing data: averages, spread and distributions

Before any test or model, an analyst describes the data: where the centre is and how spread out the values are. Screenings check that you can calculate these and choose the right one.

Mean, median and mode

• Mean: add the values and divide by the count. For 4, 6, 8 and 10 it is 28 / 4 = 7.
• Median: the middle value after sorting. With an even count, average the two middle values. For 3, 4, 5, 7, 9, 12 it is (5 + 7) / 2 = 6.
• Mode: the most frequent value. A list can have no mode or several.

The mean is pulled by extreme values. Salaries of 30, 32, 35, 38 and 200 thousand have a mean of 67 but a median of 35, so the median better describes a typical salary. In right-skewed data such as income or house prices, the mean is usually above the median.

Variance and standard deviation

Spread tells you how far values sit from the mean. For 2, 4, 4, 4, 5, 5, 7, 9:

1. Mean = 40 / 8 = 5.
2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
3. Population variance = 32 / 8 = 4. Standard deviation = square root of 4 = 2.

For a sample, divide by n - 1 instead: 32 / 7 is about 4.57, and the standard deviation is about 2.14. Questions often hide this difference in the options, so read whether the data is a full population or a sample.

The normal distribution

Many measurements, such as heights or test scores, are roughly bell-shaped. For a normal distribution:

• About 68% of values lie within 1 standard deviation of the mean.
• About 95% lie within 2 standard deviations.
• About 99.7% lie within 3 standard deviations.

With mean 70 and standard deviation 10, about 95% of scores fall between 50 and 90. A z-score expresses this distance directly: z = (value - mean) / standard deviation, so a score of 85 has z = 1.5.

Practical checklist

• Sort the data before finding a median.
• Check for outliers before reporting a mean.
• Ask whether you have a sample or a population before choosing n or n - 1.
• Always report a measure of spread next to an average.