22 min read

Recap E — Descriptive Statistics

Mean, median, mode and range from lists and frequency tables, and bar charts — 15 worked examples.

The Big Idea

Descriptive statistics summarise a set of data with a few numbers: Measure What it tells you How to find it Mean the "fair share" average add all the values, then divide by how many there are Median the middle value sort the values, then find the middle Mode the most common value the value that appears most often Range how spread out the data is largest - smallest

Measure	What it tells you	How to find it
Mean	the "fair share" average	add all the values, then divide by how many there are
Median	the middle value	sort the values, then find the middle
Mode	the most common value	the value that appears most often
Range	how spread out the data is	largest − smallest

1. From a List

Example 1 — the mean

Question: Find the mean of 6, 3, 9, 3, 4 . Answer: Total = 6 + 3 + 9 + 3 + 4 = 25 . There are 5 values. Mean = 25 \div 5 = 5

Example 2 — the median (odd number of values)

Question: Find the median of 6, 3, 9, 3, 4 . Answer: Sorted: 3, 3, 4, 6, 9 . Five values \to position \frac{5 + 1}{2} = 3 . Median = 4

Example 3 — the median (even number of values)

Question: Find the median of 12, 7, 15, 9, 7, 10 . Answer: Sorted: 7, 7, 9, 10, 12, 15 . Six values \to position 3.5, so take the mean of the 3rd and 4th values: \frac{9 + 10}{2} = 9.5

Example 4 — the mode — one, two or none

Question: Find the mode of (a) 4, 7, 7, 2, 7, 5 (b) 3, 8, 3, 6, 8, 1 (c) 2, 5, 9, 11 . Answer: (a) 7 — it appears three times (b) 3 and 8 — both appear twice (two modes) (c) no mode — every value appears once

Example 5 — the range

Question: Find the range of 18, 25, 9, 31, 14 . Answer: Largest - smallest = 31 - 9 = 22

Example 6 — all four (like the paper)

Question: 11 students' scores: 74, 58, 58, 91, 42, 63, 74, 85, 58, 67, 80 . Find the mean, median, mode and range. Answer: Sorted: 42, 58, 58, 58, 63, 67, 74, 74, 80, 85, 91 Mean = 750 \div 11 \approx 68.18 Median = 6th value = 67 Mode = 58 (three times) Range = 91 - 42 = 49

Example 7 — a missing value

Question: The mean of 5 numbers is 20 . Four of them are 18, 25, 14 and 22 . Find the fifth. Answer: Total of all 5 = 5 \times 20 = 100 . Total of the four = 79 . Fifth = 100 - 79 = 21

2. From a Frequency Table

Example 1 — make a frequency table

Question: 20 students said how many pets they own: 1, 0, 2, 1, 1, 3, 0, 1, 2, 1, 4, 2, 0, 1, 2, 1, 3, 0, 1, 2 . Make a frequency table. Answer: Count how many times each value appears (a tally helps), then add the f \times x column: Pets (x) Tally count (f) f \times x 0401882510326414 Total 2028 Check: the frequencies add up to 20 ✓

Pets ( $x$ )	Tally count (f)	f × $x$
$0$	$4$	$0$
$1$	$8$	$8$
$2$	$5$	$10$
$3$	$2$	$6$
$4$	$1$	$4$
Total	$20$	$28$

Example 2 — the mean from a table

Question: Using the pets table, find the mean. Answer: Mean = \frac{\sum f x}{\sum f} = \frac{28}{20} = 1.4 pets

Example 3 — the median from a table

Question: Using the pets table, find the median. Answer: 20 values \to the median is the mean of the 10th and 11th values. Count down: 0 pets covers positions 1 - 4; 1 pet covers positions 5 - 12 . The 10th and 11th are both 1, so the median is 1 .

Example 4 — the mode and range from a table

Question: Using the pets table, find the mode and the range. Answer: Mode: the highest frequency is 8, for 1 pet Range = 4 - 0 = 4

Example 5 — draw a bar chart

Question: Draw a bar chart of the pets data. Answer: Values (0 - 4) on the horizontal axis, frequency on the vertical axis, a title, labelled axes, and equal gaps between the bars (the data is separate whole numbers).

Bar chart titled Pets owned by 20 students: 0 pets 4 students, 1 pet 8, 2 pets 5, 3 pets 2, 4 pets 1. — $The bar chart for the pets data.$

3. Choosing and Comparing

Example 1 — an outlier

Question: Incomes (LKR): 30000, 32000, 35000, 31000, 400000 . Which average best describes them? Answer: Mean = 528000 \div 5 = 105600 — higher than four of the five incomes. Median = 32000 . The median is better, because the outlier (400000) pulls the mean up.

Example 2 — adding the same number

Question: Every value in a data set is increased by 5 . What happens to the mean and the range? Answer: The mean increases by 5 . The range stays the same, because the largest and smallest both move up by 5 .

Your turn — try these on paper, then check the answers below. 1 . Find the mean, median, mode and range of 8, 3, 5, 8, 6 . 2 . Find the median of 4, 9, 2, 7 . 3 . A table shows x = 0, 1, 2 with frequencies 3, 5, 2 . Find the mean.

Example 1 — Answers to Your Turn

1 . Sorted 3, 5, 6, 8, 8 . Mean = 30 \div 5 = 6; median 6; mode 8; range 5 2 . Sorted 2, 4, 7, 9 \to \frac{4 + 7}{2} = 5.5 3 . \sum f x = 0 + 5 + 4 = 9; \sum f = 10; mean = 0.9

Next: Recap F — Probability

More lessons in Recap and Practice · Next: Recap F — Probability · Previous: Recap D — Trigonometry