Measuring Data

Median and Mode #

The middle one, and the most common one.

The median and mode are averages like the mean: the middle one, and the most common one

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One dot per value, so repeats stack. 5 has the tallest stack, so 5 is the mode.

There are five different values. Count in from both ends and you meet at 6, the median.

With six values, two share the middle: 6 and 8. The median is halfway between: 7.

Measures of Central Tendency #

One extreme drags the mean, not the median.

One extreme value drags the mean but leaves the median where it was

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2, 4 and 6 balance at 4: the 2 pulls left as hard as the 6 pulls right.

Swap the 6 for a 24 and the mean moves to 10, but the median is still 4.

Averages from a Frequency Table #

Weight each value by how often it happened.

A frequency table averages by weighting each value with how often it happened

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The table records ten games: 1 goal in three of them, 2 goals in five, and 3 goals in two.

The mean weights each value by its frequency: 19 goals over 10 games is 1.9.

The largest frequency is 5, on the 2-goals row — so the mode is 2 goals.

The median is at position (10 + 1) ÷ 2 = 5.5. Counting down, games 5 and 6 both scored 2, so the median is 2.

Mean of Grouped Data #

Estimate using each interval midpoint.

With grouped data you estimate using the midpoint of each interval

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You no longer have the raw values, only how many fell in each interval.

Treat every value as its interval midpoint, then average as usual.

Here that is 5 × 4 + 15 × 6 = 110, and there are 10 values, so the estimated mean is 11.

The Modal Class #

The interval with the most values in it.

The modal class is the interval holding the most values, and the answer is the interval

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Grouped data do not show the individual values, so no single value can be named as the mode.

The largest frequency is 12, so the modal class is the interval 20–29 cm.

Name the interval, not its frequency — the count is the reason, not the answer.

With unequal widths, compare frequency density: 14 spread over a double width is only 7 per width of ten.

Range and IQR #

One uses the extremes, one ignores them.

The range uses the extremes and the interquartile range ignores them

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Quartiles cut the sorted values into four parts: 14 is the lower quartile, 22 the upper.

The range is 30 − 2 = 28. The middle half, 22 − 14 = 8, is the interquartile range.

Move the 30 out to 90 and it becomes an outlier, a value far from the rest.

The range jumps from 28 to 88, but the interquartile range is still 8.

An outlier lies more than 1.5 × IQR past a quartile: 22 + 1.5 × 8 = 34, and 90 is past it.

Measures of Spread #

Same average, nothing alike.

Two sets can share an average and still be nothing alike

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These three bars are all the same height. The average is 4.

These average 4 too, but they are spread far wider.

The simplest measure of spread is the range: highest minus lowest, 7 − 1 = 6.

Cumulative Frequency #

How many are below, not how many are at.

A cumulative total answers how many are below a value rather than at it

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On their own, the intervals hold 3, 8 and 6.

Add as you go and the curve never falls: 3, then 11, then 17.

Quartiles from the Curve #

Read across at a quarter, then down.

Reading across at a quarter of the total finds the lower quartile

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There are 40 people altogether, so the curve rises from 0 to 40 on the vertical axis.

A quarter of 40 is 10. Read across at 10 to the curve, then down to the axis: that is the lower quartile.

Half of 40 is 20, and reading across at 20 gives the median.

Percentiles #

The same reading, cut into hundredths.

A percentile is the value with that percent of the data below it

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Half of these 200 is 100. Read across at 100 and down: that is the median.

Percentiles divide the data into hundredths: 90% of 200 is 180, and reading across at 180 gives the 90th percentile.

A quarter of 200 is 50, which is also 25% of 200, so reading at 50 gives the 25th percentile.

So the lower quartile is the 25th percentile, the median the 50th and the upper quartile the 75th.

Box and Whisker #

The middle half drawn as a box.

A box plot draws the middle half of the data as a box

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It is drawn from five numbers: the lowest value, lower quartile, median, upper quartile and highest value.

A narrow box means the middle half of the data is close together.

Judging Normality from a Box Plot #

Symmetry keeps it open; a lean rules it out.

A box plot symmetric about its median is consistent with a normal shape, and a skewed one rules it out

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The median sits centrally, both halves of the box match, and so do both whiskers.

Here the right half of the box is three times the left, and the right whisker is longer.

A normal curve is symmetric, so a lopsided box plot rules the normal shape out.

Evenly spread data are symmetric too, so symmetry alone can never prove normality.

Mean Absolute Deviation #

The average distance from the mean.

Mean absolute deviation is the average of the distances from the mean, signs dropped

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Here are four values, and the dashed line is their mean: (1 + 3 + 6 + 10) ÷ 4 = 5.

Keep the signs and the differences from 5 are −4, −2, +1 and +5. They add to 0: above and below the mean cancel.

Their distances from 5 are 4, 2, 1 and 5. A distance carries no sign.

Average those distances: 12 ÷ 4 = 3. That is the mean absolute deviation.

These values have the same mean, 5, but sit closer to it: the distances 1, 0, 0 and 1 average to 0.5.

Standard Deviation #

How far values sit from their mean.

Standard deviation measures how far values sit from their mean on average

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The values 4, 5 and 6 sit close to their mean, 5, so the standard deviation is small.

1, 5 and 9 have the same mean but are spread far wider, so the standard deviation is larger.

Standard Deviation: Ungrouped #

Square the distances so they cannot cancel.

Square each distance from the mean before averaging so they cannot cancel

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The mean of 2, 4 and 6 is 4, so the differences from the mean are −2, 0 and +2.

Added as they are, they cancel to 0. Squared, they are 4, 0 and 4, which cannot cancel.

Average those squares, then take the square root. That is the standard deviation.

Standard Deviation: Grouped #

Same method, midpoints standing in.

Grouped data uses the same method with midpoints standing in for values

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Here are ten values in three groups, and the middle interval holds most of them.

Use the midpoints 5, 15 and 25 as the values, each weighted by its frequency; the mean is 15.

The 2 values at midpoint 5 sit 10 from the mean, so they add $2\times 10^2=200$2 × 10² = 200.

The three intervals add to 400. Divide by the 10 values to get 40, then take the square root: about 6.3.

Comparing Data Sets #

Averages and spreads together, never alone.

Compare two sets by their averages and their spreads together

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Team A has a median of 20, and its box is narrow: its scores are consistent.

On the same scale, team B also has a median of 20, but its box is much wider.

Simpson’s Paradox #

Every group up, and the total still down.

A trend can hold in every group and reverse when the groups are combined

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Treatment A wins on mild cases, 90% to 80%, and on severe too, 40% to 30%.

The counts behind them: A saw 10 mild and 100 severe, and B the other way round.

Add each row up: A cured 49 of 110, B cured 83 of 110 — 45% against 75%.

A’s cases were mostly the severe ones, so its total is dragged toward that 40%.

Finding a Frequency from the Mean #

The mean is known; one count is not.

A given mean fixes the total, and the missing count falls out of it

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The scores 1, 2 and 3 have counts 4, f and 6, and the mean is 2.1.

The mean fixes the total: with 10 + f values in all, the total is 2.1(10 + f).

Adding the column directly gives 22 + 2f. Set the two totals equal: f = 10.