Speak Maths · University
Statistics: reading a class in numbers
Statistics turns a list of measurements into a few numbers: the middle, the spread, a guess. Hats mark guesses, bars mark averages.
Draft: written by an AI helper and not yet checked by a teacher.
The alphabet
Read each sign aloud, the way it is written under it.
- Say: x barMeans: the average of the x values
- Say: x bar equals one over n times the sum of x i, for i from 1 to nMeans: add the values up and divide by how many there are
- Say: s equals the square root of one over n minus one, times the sum of x i minus x bar, squaredMeans: the sample standard deviation: the typical distance of the values from their average
- Say: sigmaMeans: the true spread of the whole group, which we usually cannot see
- Say: y hatMeans: our prediction of y; the hat marks a guess
- Say: theta hatMeans: our estimate of the unknown true value theta, worked out from the data
- Say: e i equals y i minus y hat iMeans: the residual: the real value minus the predicted one
The story: The heights of a class
Tap a line to see its words.
- Five classmates are measured. Their heights in centimetres are 150, 154, 156, 158 and 162.
- Add them up: the sum is 780.
- Divide by 5: the average height is 156.
- Each classmate’s distance from the average, with its sign: −6, −2, 0, 2 and 6.
- Square the distances and add them: 36 + 4 + 0 + 4 + 36 = 80.
- Divide by 4, one less than 5, and take the square root: the typical spread is about 4.5.
- With no other information, the best guess for a new classmate’s height is the average, 156.
- The new classmate is 160. The guess missed by 4, less than one s.
- These five are only a sample of the whole school. The school’s true average height, θ, is unknown; our estimate is 156.
- The school’s true spread, σ, is unknown too. We estimate it by s, about 4.5.
In words this story is 743 letters long; in symbols, 156.
Read it
What does it say?
Write it
Which notation says this?
The average of n numbers is their sum divided by n.
The error of a prediction is the real value minus the predicted value.
Our estimate of the unknown true value θ, worked out from the data.
The sum of the squared distances of the values from their average.
Our prediction for the i-th case.
The true spread of the whole group, which we usually cannot see.
The average of the three values 4, 6 and 11 is 7.
The residual of the fifth case: its real value minus its predicted value.
The sample standard deviation: the root of the squared distances added up and divided by n − 1.
Our estimate of the unknown σ is s.
The average of the y values.
Our estimate came out 0.2 above the true value θ.
Build it
Tap the tiles in order. One tile does not belong.
s is the square root of one over n minus one, times the sum of the squared distances from x bar.
Theta hat is one over n times the sum of the x values.
x bar equals the sum of the x values divided by n.
e i equals y i minus y hat i.
Sigma hat equals s: we estimate the true spread by the sample spread.
y hat equals x bar: with nothing else known, predict the average.
Challenges
Creative puzzles that use every sign of this chapter. Take your time; there is no clock.
1 Five heights
Five heights are listed in the rows. A tick shows when a height meets the condition in a column. Judge each sentence about this sample.
| ✓ | ✓ | ||
| ✓ | |||
| ✓ | ✓ | ||
| ✓ | |||
| ✓ | ✓ |
A tick means the row's value satisfies the column's condition. Here n = 5.
The average is 6 and the squared distances add to 58. Dividing by n − 1 = 4 (not by 5) gives s, and only the tallest value lies further from the average than s.
2 Hats, bars and letters
Pair each line with what it says.
Tap a line, then its meaning.
Bars mark averages, hats mark guesses, and Greek letters are usually the true values we cannot see.
3 The mean, mended
This should say: the average of n numbers is their sum divided by n. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
An average divides by how many values there are: n, not n − 1. The n − 1 belongs to the sample standard deviation.
4 How far off?
This should say: the residual is the real value minus the predicted value. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
A residual measures the miss: real minus predicted. Adding the prediction would say nothing about the miss.
5 The missing height
Four values have average 6; three of them are known. What is the fourth?
Four values with average 6 sum to 24. The three known add to 16, so the missing one is 8.
6 Too high or too low?
Compute the residual. A negative number means the prediction was too high.
e_i = y_i − ŷ_i = 7 − 9 = −2: the prediction was 2 too high.
7 What the data can tell
Which of these can be worked out from the data alone?
θ and σ are the true values behind the whole group, which we cannot see. θ̂ is a guess computed from the sample.
8 Your best guess
The school's true average height is θ. Write that our estimate of it is the average of the sample, then compare.
One way:
The hat marks a guess, the bar marks an average: θ̂ = x̄ says the sample average is our estimate of the true average.
9 Five rainy days
Rainfall in millimetres on five days. A tick shows when a day meets the condition in a column. Judge each sentence about this sample.
| ✓ | |||
| ✓ | |||
| ✓ | |||
| ✓ | ✓ |
A tick means the row’s value satisfies the column’s condition. Here n = 5.
The total is 20, so x̄ = 4 and the signed distances −4, −2, 0, 0, 6 add to zero, as they always do. Their squares add to 56; dividing by n − 1 = 4 gives s = √14 ≈ 3.7, and only the dry day and the wet day lie further than that from the average.
10 Four forecasts
Four days: the real temperature y and the forecast ŷ. A tick shows how the miss e = y − ŷ behaved. Judge each sentence.
| ✓ | ✓ | ||
| ✓ | |||
| ✓ | |||
A tick means the row’s day satisfies the column’s condition, with e_i = y_i − ŷ_i.
Real minus forecast: 2, −2, 0, −1. The second miss is −2, not 2: the forecast was too high. A perfect forecast gives e = 0, not e = y. The misses add to −1 and the real values average 34/4 = 8.5.
11 Which number is it?
Five quantities from one sample. Pair each with what it is.
Tap a line, then its meaning.
The total divided by the count is the average; s² is the sample variance, the spread squared; θ̂ − θ is how far the estimate missed the truth, which we usually cannot know.
12 Distance from what?
This should be the sample variance: the squared distances of the values from their own average, added and divided by n − 1. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
Spread is measured from the sample’s own average x̄, not from a prediction ŷ. Squaring stops the plus and minus distances cancelling.
13 Hat and bar
This should say: our estimate of the true value θ is the sample average. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
The hat marks a guess, the bar marks an average. The average of the sample is x̄; there is no such thing as a guess of the data itself.
14 Two cups of tea
Two cups cooled for 3 and 7 minutes. What is the sample standard deviation?
The average is 5; the distances are −2 and 2, squares 4 and 4, sum 8. Divide by n − 1 = 1 and take the root: √8 = 2√2 ≈ 2.8, a little more than the distance 2.
15 Everyone grew
Each plant in a tray was 10 cm tall on average. A month later every plant is 5 cm taller. What is the new average?
Adding 5 to every value adds 5 to the sum of n values, that is 5n, and so 5 to the average: 15. Multiplying would be a different story.
16 The bus forecast
The timetable predicted 20 minutes; the bus took 23. What is the residual, and was the forecast too high or too low?
e = y − ŷ = 23 − 20 = 3: positive, so the real value was above the forecast, which was too low.
17 The sample and the truth
A sample’s spread came out 4.1. Which line can honestly be written about the true spread σ of the whole group?
We never see σ itself; the sample gives an estimate, so the hat is needed: σ̂ = 4.1. Writing σ = 4.1 claims to know the truth, and squaring would give a variance, not a spread.
18 The misses cancel
When the prediction is the average, the misses balance: the residuals add up to zero. Write it in symbols, then compare.
One way:
Each residual is y_i − ȳ, and the sum of the y_i equals n times ȳ, so the positive and negative misses cancel exactly.
19 An estimate is not the truth
The spread of the sample is our estimate of the true spread, but it is not the true spread itself. Write it in symbols, then compare.
One way:
σ̂ = s says what we use as our guess; the hat keeps honest the fact that σ, the real spread of the whole group, stays unseen.
Chapter finished. The symbols are yours.
Tell your own
Measure something about five things around you: lengths, minutes, steps. Tell it in two or three lines: the values, x bar, and a guess with a hat.
Kept on this device only.