Centre & Spread — Print

One number for the middle is never enough, because two very different groups can share an average.

What the practice questions cover

These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Two classes both average 80 on a test.

Class A scored 78, 80, 80, 81, 82. Class B scored 60, 70, 80, 90, 100.

same average, very different classes Class A 60 70 80 90 100 Class B 60 70 80 90 100 both average 80 the centre alone does not describe a class

The averages are identical. The classes are not remotely alike.

In one, everyone understood about the same amount. In the other, some sailed through and some were lost.

A teacher would treat those classes completely differently, and the mean cannot tell them apart. That is why spread has to be reported alongside the centre.

A box plot and the five-number summary the box holds the middle half; the whiskers show the rest min 4 Q1 9 median 14 Q3 20 max 27 IQR = 20 - 9 = 11 range 23 051015202530 each of the four pieces holds about a quarter of the values

Three kinds of middle

The mean is the total shared out evenly. The median is the middle one in order. The mode is the commonest.

For 3, 4, 4, 5, 14: mean 6, median 4, mode 4.

They can differ sharply, and which is most useful depends on the data.

Order before you take a median

The median of an unsorted list is meaningless. Sorting first is not optional, and it is where most median errors come from.

Which centre to use

The mean uses every value, which makes it sensitive to extremes.

The median ignores how far out the extremes are, so it is steadier when there are outliers.

For 12, 15, 15, 18, 140 the median of 15 describes a typical value far better than the mean of 40.

One big value

A single outlier can move the mean a long way and the median hardly at all. That difference is a signal worth noticing.

Range: the quickest measure of spread

Range is largest minus smallest — one subtraction.

It is easy but fragile, because it depends on only two values.

One unusual value can make a tightly clustered set look widely spread.

Two values decide it

A class of scores all near 80 with one at 30 has a range of over 50. The range says nothing about the ninety-eight per cent in the middle.

Mean absolute deviation

Find the mean, then find how far each value sits from it, then average those distances.

Distances are taken as positive, since a value five below is just as far out as one five above.

A small MAD means everyone is close to the middle; a large one means they are scattered.

Why the distances are positive

Signed differences always add to zero — that is what the mean does. Taking them as distances is what stops the answer being zero every time.

Reporting honestly

Give a centre and a spread together. Either one alone can mislead.

"Mean 80, MAD 1.2" and "mean 80, MAD 12" describe completely different classes.

Adding the shape makes the description complete.

Two numbers, not one

A single average is the most common way to mislead with true numbers. The spread is what stops it.

Worked examples

Mean and median

Find the mean and median of 3, 4, 4, 5, 14.

  1. Mean: 3+4+4+5+145 = 305 = 6. Total divided by the count.
  2. Already in order, so the median is the third value: 4. Five values, so the middle is the third.
  3. The 14 pulls the mean above every other value but one. A single outlier moving the mean.

Mean 6, median 4

Choosing a centre

House prices in a street are $210k, $220k, $230k, $240k and $1.2m. Which centre describes a typical house?

  1. The mean is 210+220+230+240+12005 = 420 thousand. Dragged up by the mansion.
  2. No ordinary house on the street costs anything like $420k. The mean describes none of them.
  3. The median is $230k, which is typical. The median resists a single extreme value.

The median, $230k

Mean absolute deviation

Find the MAD of 4, 8, 10, 14.

  1. Mean: 364 = 9. Needed before any distances.
  2. Distances from 9: 5, 1, 1, 5. All taken as positive.
  3. Average them: 124 = 3. A typical value sits 3 from the mean.

3

Words to know

Mean
The total divided by how many values there are.
Median
The middle value once they are in order.
Mode
The value that appears most often.
Range
The largest value minus the smallest.
Mean absolute deviation
The average distance of the values from the mean.

Mistakes to watch for

Taking the median without sorting. The median of 5, 1, 9 is 5, not 1 — the list must be in order before the middle means anything.

Using the mean when one value is extreme. House prices of $210k to $1.2m have a mean of $420k, which describes no house on the street. The median of $230k does.

Reporting a centre with no spread. "Mean 80" fits a class where everyone scored 80 and a class split between 60 and 100. The spread is what separates them.

Letting the deviations keep their signs. Signed differences from the mean always add to zero, so the MAD would be zero every time. Take each distance as positive.

Trusting the range on a small set. It depends on only the two extreme values, so one outlier makes a tightly grouped set look wildly spread.

The mean is the balance point: if the values were weights on a ruler, it is where the ruler would balance.

Median income is usually reported rather than mean income, because a few very large incomes drag the mean well above what most people earn.

The mean absolute deviation asks a simple question — how far is a typical value from the middle — and answers it in the original units.

Problem 1

Find the mean of 4, 8, 10, 14.

Worked answer

9

36 ÷ 4 = 9

Problem 2

Find the mean of 3, 7, 8.

Worked answer

6

18 ÷ 3 = 6

Problem 3

Find the median of 3, 7, 9, 12, 15.

Worked answer

9

Five values, so the third

Problem 4

Find the mode of 2, 3, 3, 5, 7, 3.

Worked answer

3

3 appears three times

Problem 5

Find the range of 12, 19, 4, 15.

Worked answer

15

19 - 4 = 15

Problem 6

Find the median of 5, 1, 9, 3.

Worked answer

4

1,3,5,9

3+52 = 4

Problem 7

Find the median of 8, 2, 11, 5, 7.

Worked answer

7

2,5,7,8,11

The third is 7

Problem 8

Find the mean of 3, 4, 4, 5, 14.

Worked answer

6

30 ÷ 5 = 6

Problem 9

Find the mean absolute deviation of 4, 8, 10, 14.

Worked answer

3

Mean 9

Distances 5,1,1,5

12 ÷ 4 = 3

Problem 10

The mean of five numbers is 12. Find their total.

Worked answer

60

12 × 5 = 60

Problem 11

Prices are $210k, $220k, $230k, $240k and $1.2m. Which describes a typical house better?

  1. The median
  2. The mean
  3. The range
  4. The mode
Worked answer

The median

Mean $420k describes no house on the street

Problem 12

Find the mean absolute deviation of 2, 4, 6, 8.

Worked answer

2

Distances 3,1,1,3

8 ÷ 4 = 2

Problem 13

Four numbers have mean 7. Three of them are 5, 6 and 9. Find the fourth.

Worked answer

8

28 - 20 = 8

Problem 14

Two classes both average 80. Class A has MAD 1.2, class B has MAD 12. Which is true?

  1. Class B is far more spread out
  2. They are identical
  3. Class A is more spread out
  4. Cannot tell
Worked answer

Class B is far more spread out

Class B has students far from the middle

Problem 15

Find the median of 4, 4, 7, 9, 10, 12.

Worked answer

8

7+92 = 8

Problem 16

Adding one very large value to a data set will

  1. raise the mean much more than the median
  2. raise the median more
  3. change neither
  4. lower both
Worked answer

raise the mean much more than the median

The mean does; the median barely moves

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