Egypt Preparatory 1 Mathematics, Week 9: The arithmetic mean from a frequency table

Arithmetic mean, median and mode

Week 9 · Arithmetic mean, median and mode: from lists and frequency tables, and misleading statistics · 3 lessons

The arithmetic mean from a frequency table

Mathematics Prep 1 Term 1, Unit 3, Lesson 2

GoalFind the mean from a frequency table and find a missing frequency for a given mean.

New words

frequency table · a table of values x and how many times f each occursf × x · a value times its frequency: its share of the totalΣ f · the sum of the frequencies: the number of datamissing frequency · an unknown f found from the mean

Explanation

When the data comes as a frequency table, each value x occurs f times, so it adds f × x to the total. Add a column f × x. Then mean = Σ f·x ÷ Σ f, where Σ means “the sum of”. Divide by the sum of the frequencies, which is the number of data, not by the number of rows. If one frequency is missing and the mean is given, write Σ f·x and Σ f using k, then solve the equation Σ f·x = mean × Σ f.

ƒFormula

mean = Σ f·x ÷ Σ f

Divide by the total frequency, not by the number of rows.

Worked examples

Books read x	Pupils f	f × x
0	3	0
1	5	5
2	6	12
3	4	12
4	2	8
Total	20	37

Twenty pupils and the books they read in a month. Σ f = 3 + 5 + 6 + 4 + 2 = 20. Σ f·x = 0 + 5 + 12 + 12 + 8 = 37. Mean = 37 ÷ 20 = 1.85 books.

A missing frequency. x = 1, 2, 3, 4 and f = 2, 5, k, 3, and the mean is 2.6. Σ f = 10 + k and Σ f·x = 2 + 10 + 3k + 12 = 24 + 3k. So 24 + 3k = 2.6 × (10 + k) = 26 + 2.6k. Then 0.4k = 2, so k = 5.

x	f	f × x
1	2	2
2	5	10
3	5	15
4	3	12
Total	15	39

Check with k = 5. The table is complete. Σ f = 15 and Σ f·x = 39. Mean = 39 ÷ 15 = 2.6, as given. A common mistake is to divide 39 by 4, the number of rows: that gives 9.75, which is bigger than every value.

Class activity

Dice means. With a grown-up, roll a die 20 times. Make a frequency table of the numbers rolled with a column f × x. Work out the mean. Is it close to 3.5?

Practice

x = 1, 2, 3 and f = 4, 3, 3. Find the mean.

x = 10, 20, 30 and f = 2, 5, 3. Find the mean.

x = 0, 1, 2, 3 and f = 5, 3, k, 2. The mean is 1. Find k.

Goals in 10 matches: 0 goals in 3 matches, 1 goal in 4, 2 goals in 3. Find the mean number of goals.

Nour divides Σ f·x = 19 by the 3 rows of the table x = 1, 2, 3 (f = 4, 3, 3). What is wrong?