A value and how often it happened
A team plays 10 games. It scores 1 goal in 3 of them, 2 goals in 5 of them and 3 goals in 2 of them. Written out in full, the list of goals is 1, 1, 1, 2, 2, 2, 2, 2, 3, 3.
A frequency table says the same thing in three rows. Each row gives a value, here a number of goals, and its frequency, the number of games in which that value happened. Every game is counted in exactly one row, so the frequencies add up to the number of games: 3 + 5 + 2 = 10.
The frequency column counts games: 3, 5 and 2, which make 10 games in all.
The mean: the total of the values, shared out
The mean is the total of all the values divided by how many values there are. Here that is the total number of goals divided by the number of games.
The total number of goals comes from the rows. The team scored 1 goal in each of 3 games, which is 1 × 3 = 3 goals. It scored 2 goals in each of 5 games, which is 2 × 5 = 10 goals, and 3 goals in each of 2 games, which is 3 × 2 = 6 goals. Altogether that is 3 + 10 + 6 = 19 goals, the same as adding the ten values in the long list one by one.
Now share the 19 goals out over the 10 games: 19 ÷ 10 = 1.9 goals a game.
In symbols, write x for a value and f for its frequency. The sign , the Greek capital letter sigma, means “add up”. Each row gives the product f × x, the total of the values in that row, and the mean is : the total of the values divided by the total frequency.
Each row’s goals scored is its number of goals times its number of games. The totals, 10 games and 19 goals, give the mean 19 ÷ 10 = 1.9.
Divide by the games, not by the rows
The usual mistake is to divide by the number of rows: goals a game. The table has 3 rows, but those rows describe 10 games, and the mean is a mean for each game. A mean of 6.3 goals cannot be right, because no game had more than 3 goals, and a mean always lies between the smallest and the largest value.
The other mistake is to add the frequency column and treat that as the total: 3 + 5 + 2 = 10 is the number of games, not the number of goals. The total of the values needs each value times its frequency.
The mode: the value with the largest frequency
The mode is the value that happened most often. In a frequency table, that is the value in the row with the largest frequency. The largest frequency is 5, in the row for 2 goals, so the mode is 2 goals.
The mode is 2 goals, not 5. The 5 is how many games had 2 goals; it is the reason 2 goals is the mode, and the answer is the value beside it.
The row for 2 goals has the largest frequency, 5, so the mode is 2 goals.
The median: count along a running total
The median is the middle value of the ordered list. With n values it sits in position . Here n = 10, and , so the median is halfway between the 5th and the 6th game in order.
The table is already in order, from 1 goal to 3 goals, so count down the frequency column keeping a running total. The first row holds the 1st to the 3rd games. The second row holds the next 5, so the running total reaches 3 + 5 = 8: the 4th to the 8th games all had 2 goals. The 5th and the 6th games are both in that row, so the median is 2 goals.
The median is found by counting games, not by picking the middle row of the table. A second team scored 0 goals in 6 games, 1 goal in 2 games, 2 goals in 1 game and 3 goals in 1 game. The middle rows of its table are 1 and 2 goals, but the running total is already 6 after the first row, so the 5th and the 6th games both had 0 goals and the median is 0 goals.
The running total goes from 3 to 8 in the row for 2 goals, so that row holds the 4th to the 8th games, the 5th and the 6th among them. The median is 2 goals.
Worked example: Goals in a Season Read from a Frequency Table: the Mean, the Median and the Mode
Question A hockey team played 20 matches in a season. It scored 0 goals in 3 matches, 1 goal in 5 matches, 2 goals in 6 matches, 3 goals in 3 matches, 4 goals in 2 matches and 5 goals in 1 match. (a) Find the mean number of goals the team scored in a match. (b) Find the median and the mode, and say in how many matches the team scored more than the mean.
1.Write the table in three columns: the goals x, the number of matches f, and the product fx, which is the number of goals scored in the matches of that row.
The table is set out in three columns: the goals x, the matches f, and the product fx. 2.Fill in the fx column: 0 × 3 = 0, 1 × 5 = 5, 2 × 6 = 12, 3 × 3 = 9, 4 × 2 = 8 and 5 × 1 = 5.
Each row of the fx column is the goals scored in the matches of that row: 0 × 3 = 0, 1 × 5 = 5, and so on. 3.(a) Total the two columns: 3 + 5 + 6 + 3 + 2 + 1 = 20 matches and 0 + 5 + 12 + 9 + 8 + 5 = 39 goals. The mean is 39 ÷ 20 = 1.95 goals in a match.
(a) The totals are 20 matches and 39 goals, so the mean is 39 ÷ 20 = 1.95 goals in a match. 4.For the median, 20 values put it halfway between the 10th and the 11th. Counting along the frequencies, 3 matches had 0 goals and 3 + 5 = 8 had 1 or fewer, so the 9th to the 14th matches all had 2 goals. The 10th and the 11th are both 2, so the median is 2 goals. The largest frequency is 6, at 2 goals, so the mode is 2 goals as well.
Counting along the frequencies, the 9th to the 14th matches all had 2 goals, so the median is 2. The largest frequency is 6, at 2 goals, so the mode is 2. 5.(b) More than 1.95 goals means 2 goals or more, which is 6 + 3 + 2 + 1 = 12 matches. The median is 2 goals, the mode is 2 goals and 12 of the 20 matches were above the mean. Check: 8 matches were below the mean and 8 + 12 = 20.
(b) The matches above the mean are those with 2 goals or more: 6 + 3 + 2 + 1 = 12 of the 20.
Answer: (a) the team scored 39 goals in the 20 matches, so the mean is 1.95 goals in a match; (b) the median is 2 goals, the mode is 2 goals, and 12 of the 20 matches were above the mean
Common mistakes
- Dividing the total number of goals by the number of different scores instead of by the number of matches: 39 ÷ 6 = 6.5. There are 6 rows in the table but 20 matches, and the mean is a mean for each match, so the divisor is the total of the f column.
- Reading the mode as 6, the largest number in the f column. That 6 is how often the mode happened, not the mode itself. The mode is the value that happened most often, which is the x beside that frequency, 2 goals.