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Σfx ÷ Σf

Frequency table calculator

Values with their counts produce the weighted mean, variance, standard deviation, median and mode — the layout exam questions actually use.

Short answer: the table x = 1, 2, 3, 4 with f = 3, 5, 2, 1 has 11 observations, a mean of 2.0909, a mode of 2 and a median of 2. The mean is 23 ÷ 11, not 2.5, because the value 2 appears five times.

Why counts matter

Averaging the four values gives 2.5; weighting them gives 2.09. On a 100-unit question that 0.4 difference is the whole mark, and it comes from treating the frequency as a weight rather than a separate column to ignore.

Worked example

Test scores of 2, 4, 4, 4, 5, 5, 7, 9 have mean 5 and a population standard deviation of exactly 2. That is the same answer the statistics calculator gives when the values are typed out one by one, which is the check that the weighting is right.

Frequency table questions

How is a frequency table different from a list?

Repeated values are counted rather than written out: x = 1, 2, 3, 4 with f = 3, 5, 2, 1 means the number 1 appears three times, so there are 11 observations, not 4. The mean is Σfx ÷ Σf.

Population or sample standard deviation?

Population, dividing by Σf, because a frequency table is normally the whole data set. If your data is a sample from a larger group, the sample divisor is Σf − 1 and the figure here will read slightly low — the page states which convention it uses rather than hiding it.

How are the median and mode found here?

By expanding the counts — which is exact and fast at these sizes — and then taking the middle value and the value with the highest frequency. The median of an even count is the average of the two middle values.

Can frequencies be decimal?

No. A frequency is a count, so a fractional one is rejected instead of being quietly multiplied through. Weighted data with fractional weights belongs in the statistics calculator, which says so on its own page.

Related tools

Statistics calculatorMean, median, quartiles and SD from a list.Percentage calculatorShare of total and change.Rounding calculatorRound the figures to the marks asked for.