Statistics

Study Sheet

Statistics

Mean, median and spread

Two Kinds of Middle

Concept
The Big Idea

The mean shares the total out equally. The median is the value in the middle once the data is in order. One outlier drags the mean but barely moves the median.

The mean is the fair-share average: total divided by count. The median is the middle value when the data are sorted, and the mode is the most common value. The mean is pulled by extreme values; the median is not, which is why house prices are reported as medians.

Tip
Even-sized sets

With an even count there is no single middle value, so the median is the average of the two middle numbers.

Example
Mean and median of 2,3,3,5,122, 3, 3, 5, 12
012345678910111213median 3mean 5
One large value drags the mean to the right of most of the data.

Mean: the total 2+3+3+5+12=252 + 3 + 3 + 5 + 12 = 25, shared among 55 values, is 25÷5=525 \div 5 = 5. Median: the data are in order, and the middle (third) value is 33. The 1212 pulls the mean up to 55, higher than three of the five values; the median stays at 33.

Example
An even count: 4,7,9,124, 7, 9, 12

There are two middle values, 77 and 99. The median is halfway between them: (7+9)÷2=8(7 + 9) \div 2 = 8.

Side note
Where it goes wrong

Forgetting to sort. For the data 9,2,79, 2, 7 the middle of the list as written is 22, but in order the data are 2,7,92, 7, 9 and the median is 77. Also, average the two middle values only when the count is even; with an odd count there is a single middle value.

Try it
Try it: mean and median

Find the mean and the median of 6,1,8,56, 1, 8, 5.

Show the answer

Mean: (6+1+8+5)÷4=20÷4=5(6 + 1 + 8 + 5) \div 4 = 20 \div 4 = \mathbf{5}. Median: in order, 1,5,6,81, 5, 6, 8; the middle two are 55 and 66, so the median is 5.5\mathbf{5.5}.

In Plain Terms & More Examples

Concept
In plain terms

The mean shares the total out equally; the median is the middle value once the numbers are in order.

Example
Mean of 4,6,8,104, 6, 8, 10
0246810#1#2#3#4mean 7
Level the bars: what the tall ones have above 7 fills the short ones up to 7.

Total =28= 28, and 28÷4=728 \div 4 = 7, so the mean is 77.

Example
Median of 3,5,93, 5, 9

In order, the middle value is 55.

Side note
Where it goes wrong

Dividing by the wrong count. The mean of 4,6,8,104, 6, 8, 10 is 28÷4=728 \div 4 = 7. Dividing by 55, or any number other than how many values there are, gives 5.65.6, which is not the fair share.

Try it
Try it: one of each

Find the mean of 7,9,11,137, 9, 11, 13 and the median of 10,2,8,6,410, 2, 8, 6, 4.

Show the answer

Mean: 40÷4=1040 \div 4 = \mathbf{10}. Median: in order, 2,4,6,8,102, 4, 6, 8, 10; the middle value is 6\mathbf{6}.

Which Average? And the Range

Concept
Outliers move the mean

One extreme value drags the mean but leaves the median alone. The range (largest minus smallest) measures spread but is also hit by outliers.

Example
Salaries
05010015020030323538200mean 67median 35
The mean line sits above four of the five salaries; the median sits among them.

Five workers earn 30,32,35,3830, 32, 35, 38 and 200200 (thousands). Mean =335÷5=67= 335 \div 5 = 67; median =35= 35. The median describes a typical worker better.

Example
Range and an outlier
1015202530354045
Five values bunch together; one sits far away.

Data: 12,14,15,15,16,4112, 14, 15, 15, 16, 41. Range =41−12=29= 41 - 12 = 29. Without the 4141, the range is 16−12=416 - 12 = 4. One unusual value made the spread look seven times larger.

Side note
Where it goes wrong

Using the mode as "typical" without looking. In 1,1,7,8,91, 1, 7, 8, 9 the mode is 11, but most values are near 88. The median, 77, describes this data much better.

Try it
Try it: compute all four

Data: 8,3,5,8,68, 3, 5, 8, 6. Find the mean, median, mode and range.

Show the answer

Sorted: 3,5,6,8,83, 5, 6, 8, 8. Mean 30÷5=630 \div 5 = \mathbf{6}; median 6\mathbf{6}; mode 8\mathbf{8}; range 8−3=58 - 3 = \mathbf{5}.

Working Backwards

Concept
Mean times count is the total

If nn values have mean mm, their total is n×mn \times m. Working backwards from a mean always starts by finding this total.

The mean is the fair share, so nn fair shares rebuild the total. With the total in hand you can find a missing value (the total minus the known values) or a new mean (the new total divided by the new count).

Example
The missing score
020406080100809088?mean 85
The fourth bar must make the four bars level out at 85.

Four tests average 8585, so the total is 340340. Three scores are 80,90,8880, 90, 88 — that is 258258 — so the fourth is 340−258=82340 - 258 = 82.

Example
Adding a value

Five quiz scores have mean 8080, so their total is 5×80=4005 \times 80 = 400. A sixth score of 9292 makes the total 492492 and the mean 492÷6=82492 \div 6 = 82.

Side note
Where it goes wrong

Averaging the averages. If a class of 1010 averages 8080 and a class of 3030 averages 6060, the combined mean is not (80+60)÷2=70(80 + 60) \div 2 = 70. The totals are 800+1800=2600800 + 1800 = 2600 for 4040 students, a mean of 6565. The bigger class counts for more.

Try it
Try it: what do I need?

Priya has scores 72,85,9172, 85, 91. What must she score on a fourth test to average exactly 8585?

Show the answer

Needed total 4×85=3404\times85 = 340. Current total 248248. She needs 340−248=92340 - 248 = \mathbf{92}.

Dot Plots and Histograms

Concept
Every value, or grouped values

A dot plot shows every value. A histogram groups values into equal intervals and draws a bar for how many fall in each; the bars touch because the intervals do.

With a handful of values a dot plot keeps every detail. With many values spread widely, single dots become clutter, and grouping them into intervals shows the overall shape: where the data bunch up, and whether they trail off to one side.

Example
Pets
01234567number of pets
Twelve dots, one per student.

Twelve students count their pets: 0,0,0,1,1,1,1,2,2,3,4,60, 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 6. The mode is 11. With 1212 values the median is the mean of the 66th and 77th, which are both 11: median 11. The mean is 21÷12=1.7521 \div 12 = 1.75, pulled up by the student with 66.

Example
Reading minutes
024680–910–1920–2930–39minutes
Touching bars: each covers an interval of 10 minutes.

A histogram shows how long 1616 students read: 00 to 99 minutes, 33 students; 1010 to 1919, 77; 2020 to 2929, 55; 3030 to 3939, 11. How many read at least 2020 minutes? 5+1=65 + 1 = 6 students.

Side note
Where it goes wrong

Reading a bar as one value. The 1010-to-1919 bar of height 77 means seven students read somewhere from 1010 to 1919 minutes, not that seven read exactly 1010. A histogram hides the exact values, so it cannot give an exact mean.

Try it
Try it: the pets plot

From the pets data, how many students have more than 22 pets, and what fraction of the class is that?

Show the answer

More than 22: the values 33, 44 and 66, so 3\mathbf{3} students. That is 312=14\tfrac{3}{12} = \mathbf{\tfrac14} of the class.

Why It Works

The mean is the value every data point would have if the total were shared equally: nn copies of the mean add to the same total as the data. That is why the deviations from the mean always sum to zero — the amounts above balance the amounts below. The median splits the sorted list into two equal halves by count, so it does not care how far away the largest value is; that is its robustness.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.