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

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

Five different values. Walk 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

Two, four and six balance at four: the 2 pulls left as hard as the 6 pulls right.

Swap the 6 for a 24 and the pivot slides to 10 — but the middle value 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

Ten games in three rows: 1 goal three times, 2 goals five times, 3 goals twice.

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 sits at position $\left(n+1\right)/2$(n + 1) / 2 — count down the frequency column to reach it.

Mean of Grouped Data #

Estimate using each interval midpoint.

With grouped data you estimate using the midpoint of each interval

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 plus 15 × 6, which is 110. Ten values, so the estimate 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

Grouped data hide the individual values, so no single value can be 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 over two widths is only 7 each.

Range and IQR #

One uses the extremes, one ignores them.

The range uses the extremes and the interquartile range ignores them

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.

Drag that 30 out to 90 and it is an outlier — a value far outside the rest.

The range jumps from 28 to 88. The middle half is untouched, still 8 wide.

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

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

On their own the intervals hold three, then eight, then six.

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

Forty people altogether, so the curve climbs from 0 up to 40 on the side.

A quarter of 40 is 10. Read across at 10, stop at the curve, drop to the value below.

Halfway up is 20, and reading across there lands on the median instead.

Percentiles #

The same reading, cut into hundredths.

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

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

A percentile cuts the count into hundredths: 90% of 200 is 180, giving the 90th.

A quarter of 200 is 50, and 25 hundredths lands in the same place: the 25th.

So the lower quartile is the 25th, the median the 50th, 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

Five numbers draw it: lowest, lower quartile, median, upper quartile and highest.

A narrow box means the middle half is packed tightly 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

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

Four values, and the dashed line is their mean: 1 + 3 + 6 + 10 divided by 4 is 5.

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

Average those distances and the answer is 3. That is the mean absolute deviation.

Same mean of 5, packed tighter: distances 1, 0, 0 and 1 average to just 0.5.

Standard Deviation #

How far values sit from their mean.

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

These sit close to their mean, so the standard deviation is small.

Same mean, but 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

The distances from the mean are −2, 0 and +2.

Added as they are, they cancel to zero. Square them first and they 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

Ten values, grouped, with the middle interval holding most of them.

Use each midpoint — 5, 15, 25 — weight it by its count, then carry on as before.

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

All three come to 400. Share that over the 10 values, then take the root: 6.3.

Comparing Data Sets #

Averages and spreads together, never alone.

Compare two sets by their averages and their spreads together

One team averages 20, and its box is narrow: its scores are very consistent.

On the same scale, team B averages 20 too — but its box stretches 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

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

Scores 1, 2 and 3 with counts 4, f and 6 — and the mean is said to be 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.