🔔The Normal Distribution

Surveyor Halden keeps every measurement the Tower has ever taken, and says the same shape keeps turning up. Measure the heights of a thousand reeds, the times of a thousand ferry crossings, the weights of a thousand grain sacks, and the picture is always a bell: a heap in the middle, thinning out evenly on both sides, with no second bump and no long tail on one side only.

A bell like that is called a normal distribution, and two numbers describe the whole of it:

• ◆the mean, which sits under the peak — it is also the median, because the curve is symmetric;
• ◆the standard deviation, which says how wide the heap is.

Because the shape is always the same, the share of the data in any band is always the same too, once you measure that band in standard deviations. That is the empirical rule, and it is worth knowing by heart:

about 68% within 1 standard deviation of the mean about 95% within 2 about 99.7% within 3

Symmetry then splits each band in half, and those halves are where most of the real questions live:

mean to 1 standard deviation above 34% (half of 68) 1 to 2 standard deviations above 13.5% (half of 95 - 68) 2 to 3 standard deviations above 2.35% above 2 standard deviations 2.5% (half of 100 - 95) below 1 standard deviation under 16% (half of 100 - 68)

So for reed heights with mean 70 cm and standard deviation 8 cm: about 68% of the reeds measure between 62 and 78 cm, about 34% between 70 and 78, about 2.5% above 86, and about 16% below 62. Everything there came from the mean, the standard deviation, and one memorised line.

The empirical rule only answers questions about whole numbers of standard deviations. For 74 cm, or 83 cm, you need the machinery underneath, and it starts with a z-score:

z = (value - mean) / standard deviation

A z-score is a distance measured in standard deviations. z = 1.4 means "1.4 standard deviations above the mean"; z = -0.8 means "0.8 standard deviations below it". A z-score has no units, which is exactly what makes it useful: it turns centimetres, minutes and points into one common ruler.

Once you have z, a standard normal table turns it into an area. Every question in this skill prints the rows it needs, so you never have to remember one. It looks like this:

z Area to the left of z 0.00 0.5000 0.50 0.6915 0.80 0.7881 1.00 0.8413 1.20 0.8849 1.30 0.9032 1.40 0.9192 1.50 0.9332 1.60 0.9452 1.80 0.9641 2.00 0.9772 2.50 0.9938

Read that column carefully: it is the area to the left of z, which is the proportion of the data BELOW that value. Areas are proportions between 0 and 1, so multiply by 100 to get a percent. 0.9192 is 91.92%, which rounds to 91.9%.

Two moves cover everything else.

Above instead of below. The whole curve is 1, so the area to the right is 1 minus the area to the left. If z = 1.4, then below is 0.9192 and above is 1 - 0.9192 = 0.0808, that is 8.1%.

A negative z. Tables print positive rows only, because the curve is symmetric and the negative half is a mirror of the positive half. The area to the left of -z equals 1 minus the area to the left of +z:

area left of -1.5 = 1 - 0.9332 = 0.0668, that is 6.7%

Sketch the bell and shade the region before you look anything up. One shaded picture prevents almost every error in this topic.

A question that names two values is really two look-ups. Both table entries measure from the far left, so the strip between them is the bigger area minus the smaller one — never the sum.

P(a < X < b) = (area left of z for b) - (area left of z for a)

Take grain sack weights with mean 70 kg and standard deviation 10 kg, and ask what share weighs between 78 and 84 kg.

z for 78 = (78 - 70) / 10 = 0.8 -> 0.7881 z for 84 = (84 - 70) / 10 = 1.4 -> 0.9192 between = 0.9192 - 0.7881 = 0.1311, that is 13.1%

If the strip straddles the mean, one z is negative and you use symmetry on that one first. Between 55 and 84, with the same sacks:

z for 55 = -1.5 -> 1 - 0.9332 = 0.0668 z for 84 = 1.4 -> 0.9192 between = 0.9192 - 0.0668 = 0.8524, that is 85.2%

Notice the sense check: a strip that covers most of the middle of the bell ought to come out large, and 85.2% is large. A strip out in a tail ought to come out small. If your arithmetic disagrees with your sketch, trust the sketch and hunt for the slip.

Outside a range is the leftover: work out what lies between, then subtract from 100%. That is exactly how a tolerance question at the forge works. Bolts with mean length 10 mm and standard deviation 0.1 mm, sent back if they fall outside 9.8 to 10.2 mm, are being measured against limits 2 standard deviations either side of the mean — 95% inside, so about 5% go back.

To turn a percent into a count, multiply by the size of the batch and round only at the very end:

0.8524 x 500 = 426.2, so about 426 sacks

A percentile is an area wearing a different hat. Being at the 90th percentile means 90% of the data lies below you. So percentile questions run in whichever direction you are missing a piece.

Value to percentile. Find z, look up the area, multiply by 100. A reed of 78 cm among reeds with mean 70 and standard deviation 8 has z = 1, an area of 0.8413, and so sits at about the 84th percentile.

Percentile to value. Go the other way: find the area in the table, read the z beside it, then undo the z-score.

value = mean + z x standard deviation

For the 90th percentile, hunt for 0.9000 in the area column. It falls between the rows for 1.20 and 1.30, and the standard value used for it is z = 1.28 — where a question needs one of these in-between numbers, it tells you which z to use. With mean 70 and standard deviation 10:

value = 70 + 1.28 x 10 = 82.8

A cut-off is the same calculation in costume. "The top 10% are pinned to the survey board" means 90% lie below the cut-off, so the cut-off is the 90th percentile, 82.8. "The bottom 5% measure again" means z = -1.645, and the cut-off is 70 - 16.45 = 53.6. Watch the direction: a top slice sits above the mean and a bottom slice below it.

Two different tests. Raw scores from different distributions cannot be compared, because each test has its own mean and its own spread. z-scores can, because they are all on the same ruler.

Ines scored 82 where the mean was 70 and the standard deviation 8. Kofi scored 63 where the mean was 55 and the standard deviation 4.

Ines: z = (82 - 70) / 8 = 1.5 Kofi: z = (63 - 55) / 4 = 2.0

Kofi's raw score is nineteen points lower and his result is the stronger one: he stood 2 standard deviations above his own crowd, and Ines stood 1.5 above hers. Ranking three readings works the same way — convert all three, then order the z-scores, remembering that a negative z is below the average of that test.

Halden marks answers by what was asked for, so read the last line of the prompt before you type.

A proportion is a percent to 1 decimal place. Give 8.1, not 0.0808 and not "8.08%". The percent sign is optional — 8.1 and 8.1% are both accepted — but the rounding is not, so carry the four decimal places of the table right to the end and round once.

0.9192 -> 91.9 0.0808 -> 8.1 0.1311 -> 13.1

The empirical-rule answers are typed the same way: 68, 95, 99.7, 34, 13.5, 16, 2.5, 5.

A count is a whole number. "About how many of 500 sacks" wants 426, not 85.2 and not 426.2. Multiply first, round last.

A value is a whole number or a decimal, exactly as the prompt says. A percentile read straight off the table lands on a whole number of centimetres; a cut-off built from z = 1.28 or z = 1.645 is asked for to 1 decimal place.

An interval is a pair, smaller number first. The middle 95% of reeds with mean 70 and standard deviation 8 is typed as

(54, 86)

A comparison is a choice. Pick the option — the name of the person who did better, or the ranking — rather than typing a z-score. The z-scores are your working, not your answer.

Every question that goes beyond the 68-95-99.7 rule prints the table rows it needs right beside the prompt. There is nothing to memorise except the formula for z, the fact that the table gives the area to the left, and the symmetry step for a negative z.