Averages and Spread

An average is a single number standing in for many, and the whole topic is about what that substitution costs. Two data sets can share a mean and have nothing else in common — which is why every serious statement about data quotes a centre and a spread, and why "the average" on its own is the most quietly misleading phrase in statistics.

Which average should you use?

The mean when the data is symmetric, the median when it is skewed or has outliers, the mode when the data is categorical and there is no arithmetic to do. Measures of Central Tendency is the lesson that makes that choice deliberate rather than habitual.

The Mean shares the total out equally, and Median and Mode gives the other two: the median is the value in the middle once the data is sorted, and the mode is the value that occurs most often.

Salaries of 22, 24, 25, 26 and 180 (thousands): the mean is 55.4, the median is 25. Nobody in that office earns anything like 55.4, and the median is the honest summary.

Averages from a table

When the same values repeat, listing them all is wasteful. Averages from a Frequency Table shows the shortcut: multiply each value by its frequency, total those products, and divide by the total frequency. The median comes from a running total of frequencies, which is where the halfway item falls.

Grouped data is harder, because the individual values are gone. Mean of Grouped Data assumes every value sits at its class midpoint, which makes the answer an estimate — an honest one, and it should be labelled as such. The Modal Class makes the parallel point on the other side: with grouped data you can name the busiest class, but not a single modal value.

Why is an average not enough?

Because it says nothing about how tightly the data clusters around it, and that is often the interesting part. Measures of Spread makes the argument with two data sets that share a mean and look nothing alike.

The simplest spread is the range — largest minus smallest — and its weakness is obvious: one freak value sets it. Range and IQR introduces the fix. The interquartile range is the upper quartile minus the lower quartile, so it describes the middle half of the data and one extreme value cannot touch it.

Percentiles generalises the idea: the 90th percentile is the value that 90% of the data falls below. Quartiles are just the 25th, 50th and 75th percentiles under an older name, and percentiles are how growth charts, exam grades and response-time targets are all actually reported.

Cumulative frequency and quartiles

Cumulative Frequency is a running total of counts, and plotting it against the upper boundary of each class gives the S-shaped curve that answers "how many were below this value" for any value at all.

Quartiles from the Curve reads the quartiles off it directly: go up the vertical axis to a quarter of the total, across to the curve, and down to the value. The same move at half and three-quarters gives the median and the upper quartile.

What does a box plot show?

Five numbers, and the shape of the distribution between them. Box and Whisker draws the minimum, lower quartile, median, upper quartile and maximum, with the box holding the middle half of the data and the whiskers reaching to the extremes.

Its value is comparison. Two box plots on the same axis show at a glance which group is higher, which is more consistent, and whether either is lopsided — three questions that would take a paragraph in words.

Judging Normality from a Box Plot turns that reading into a test: in a symmetric distribution the median sits centrally in the box and the whiskers are of similar length, so a median pushed towards one end, or one long whisker, is evidence of skew. That matters later, because several statistical procedures assume symmetry before they are allowed to run.

How is standard deviation calculated, and why that way?

Find how far each value is from the mean, square those distances, average them, and take the square root. Standard Deviation builds it in that order so each step has a reason attached.

A gentler cousin comes first. Mean Absolute Deviation averages the distances without squaring, using absolute values to stop the positives and negatives cancelling. It is easier to explain and genuinely useful — and it makes the natural question sharp: if this works, why square anything?

Standard Deviation: Ungrouped works the practical formula — the mean of the squares minus the square of the mean — which gets the same answer in one pass through the data instead of two. Standard Deviation: Grouped extends it to frequency tables, weighting each squared value by its frequency and, for grouped data, using class midpoints with the same estimate caveat as the grouped mean.

How do you compare two data sets honestly?

Quote one centre and one spread from each, keep the pair consistent, and say what the numbers mean in the context of the data. Comparing Data Sets drills exactly that structure, because a comparison that lists four numbers and draws no conclusion earns nothing.

Consistency matters: median goes with interquartile range, mean goes with standard deviation. Mixing them — a median and a standard deviation — quotes a robust centre against a non-robust spread, so an outlier affects half the description.

And then there is the trap that survives every correct calculation. Simpson's Paradox shows two groups where treatment A beats treatment B in the mild cases and in the severe cases, yet loses overall — because the groups were different sizes and the severe cases went disproportionately to one treatment. Nothing in the arithmetic is wrong. The lesson is that an average over a mixed population is not a property of anyone in it.

The mistakes worth naming

• Reporting a grouped mean as exact. It is an estimate built on the midpoint assumption. Say so.
• Forgetting to sort before finding a median. The middle of an unsorted list is not the middle of the data.
• Pairing a median with a standard deviation. Median with IQR, mean with standard deviation — keep the pair robust or not, but not half of each.
• Plotting cumulative frequency at class midpoints. The running total is only true at the upper boundary.
• Comparing two groups on centre alone. Equal means with unequal spreads is the case where the spread is the whole story.
• Deleting an outlier because it is inconvenient. An outlier is a data point until there is a recorded reason to doubt it.

Where this leads next

These summaries are the vocabulary the rest of statistics is written in. The charts and displays guide covers the pictures they are read from, including the cumulative frequency curve and the histogram. Once each subject carries two measurements instead of one, the question becomes whether they move together — the scatter, correlation and regression guide. And the mean and standard deviation of a sample, used to say something about a population you did not measure, is the whole subject of the sampling and inference guide.

Learn this properly in the app

Math Challenge teaches each of these as an illustrated lesson — a drawn data set, a worked calculation and try-it problems that re-teach the exact question you missed rather than a nearby one. They sit inside a catalog of 800+ lessons running from first counting to university-entrance statistics.

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