Statistics
Grade 4-5Averages from Frequency Tables
When data is grouped into a frequency table, finding the mode, median and mean uses the same ideas as before, but each one needs to be worked out using the frequencies rather than a simple list. This lesson covers finding the mode, median and mean from a frequency table, and finding a missing frequency when the mean is already known.
What you need to know
- The mode is the value with the highest frequency.
- For the mean, multiply each value by its frequency, add those up, then divide by the total frequency.
- Find the median position with (n plus 1) over 2, then count through the frequencies.
- The total frequency is the sum of the frequency column, not the number of rows.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the mode from a frequency table
The mode is the data value with the highest frequency, not the frequency itself.
A table shows the number of pets owned by 27 households, with a frequency of 12 for 1 pet, the highest of any row. The mode is 1 pet, not 12.
Finding the median from a frequency table
Add up the frequencies to find the total, then find the middle position using (total + 1) ÷ 2. Add up the frequencies in order until this position is reached, and the median is the data value at that point.
For 27 households, the middle position is (27 + 1) ÷ 2 = 14th. Adding frequencies, 0 pets covers positions 1 to 5 and 1 pet covers positions 6 to 17, so the 14th position falls in the "1 pet" group, giving a median of 1.
Finding the mean from a frequency table
Multiply each data value by its frequency, add these products together to find Σfx, then divide by the total frequency, Σf.
For goals scored in 22 matches, Σfx = (0×4) + (1×8) + (2×6) + (3×3) + (4×1) = 33, and Σf = 4 + 8 + 6 + 3 + 1 = 22, so the mean is 33 ÷ 22 = 1.5.
Finding a missing frequency from a known mean
Write expressions for Σfx and Σf in terms of the unknown frequency, set their ratio equal to the given mean, then solve the resulting equation.
If the mean is 1.5 with frequencies 4, 6, a and 3 for values 0, 1, 2 and 3, then Σfx = 15 + 2a and Σf = 13 + a. Setting (15 + 2a) ÷ (13 + a) = 1.5 gives 15 + 2a = 1.5(13 + a), which solves to a = 9.
Worked examples
Three exam-style questions, fully solved
The table shows the number of pets owned by 27 households: 5 households with 0 pets, 12 with 1 pet, 7 with 2 pets, and 3 with 3 pets. Find the mode.
Easy- 1.Identify the highest frequency: 12, for 1 pet
Answer: 1 pet
The table shows the number of goals scored by a team in 22 matches: 4 matches with 0 goals, 8 with 1 goal, 6 with 2 goals, 3 with 3 goals, and 1 with 4 goals. Calculate the mean number of goals per match.
Medium- 1.Find Σfx: (0×4) + (1×8) + (2×6) + (3×3) + (4×1) = 33
- 2.Find Σf: 4 + 8 + 6 + 3 + 1 = 22
- 3.Divide Σfx by Σf: 33 ÷ 22
Answer: 1.5 goals per match
The mean number of pets per household is 1.5. The table shows the number of pets: 4 households with 0 pets, 6 with 1 pet, a households with 2 pets, and 3 with 3 pets, where a is unknown. Find the value of a.
Hard- 1.Write Σfx in terms of a: 0(4) + 1(6) + 2(a) + 3(3) = 15 + 2a
- 2.Write Σf in terms of a: 4 + 6 + a + 3 = 13 + a
- 3.Set the mean equation: (15 + 2a) ÷ (13 + a) = 1.5, so 15 + 2a = 1.5(13 + a)
Answer: a = 9
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A frequency table records goals per match: 0 goals (frequency 4), 1 goal (frequency 6), 2 goals (frequency 8), 3 goals (frequency 2). How many matches are there in total?
Using that table (0:4, 1:6, 2:8, 3:2), work out the modal number of goals.
Using that table, work out the total number of goals scored.
Using that table, work out the mean number of goals per match.
Using that table (0:4, 1:6, 2:8, 3:2), work out the median number of goals.
In a frequency table, a value x has a frequency of 5 and contributes 35 to Σfx. Work out x.
Avoid these
The mistakes students make most often
Confusing the mode with the frequency itself, giving the highest frequency number instead of the data value that has that frequency.
When finding the median, forgetting to use cumulative frequencies to locate the middle position, instead of just picking a value from the middle of the table.
When finding the mean, forgetting to multiply each value by its frequency before summing, and instead adding the raw data values only.
Dividing by the number of rows in the table instead of the total frequency, Σf, when calculating the mean.
When finding a missing frequency from a known mean, forgetting to include the unknown frequency in both the Σfx and Σf expressions before setting up the equation.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
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