Frequency Tables and Grouped Data

A raw list of numbers is hard to read, so we organise it into a frequency table. Take the number of pets owned by 20 students. List each different value in order down the left, tally the data by working through it once and marking a stroke for each value, then write the count as the frequency. Here 4 students have no pets, 8 have one, 5 have two, 2 have three and 1 has four. The tally marks group in fives, with the fifth stroke crossing the previous four, which makes counting quick. Then check: the frequencies must add to the number of data values, and 4 plus 8 plus 5 plus 2 plus 1 is 20 ✓. If it does not match, you missed a value or counted one twice.

Once the table is built, it answers questions quickly. How many students altogether? Add the frequency column, giving 20. What is the most common number of pets? The largest frequency, so 1 pet with 8 students. How many own no pets? Read the frequency beside 0, which is 4. How many own at least two? Add the frequencies for 2, 3 and 4, giving 8. And how many pets are there in total? That one is different — multiply each value by its frequency and add, so 0 times 4 plus 1 times 8 plus 2 times 5 plus 3 times 2 plus 4 times 1, which is 28 pets. Total pets and total students are not the same number, and confusing them is a common slip.

Continuous data like height creates a problem: nearly every value is different, so an ungrouped table would have thirty rows with a frequency of 1 in most of them, showing no pattern at all. The fix is to group the values into class intervals — 150 to 154, 155 to 159, and so on. Now four rows tell you far more than thirty did, and the shape of the data appears: most students are between 155 and 164. Aim for about five to eight intervals. Too few hides the detail, since two intervals tell you almost nothing, and too many brings back the original problem.

Class intervals have three rules. They must be the same width, so 150 to 154 and 155 to 159 and 160 to 164 all cover five values. They must not overlap — writing 150 to 155 then 155 to 160 leaves you unable to say where a height of exactly 155 belongs. And they must not leave gaps, because 150 to 154 followed by 156 to 160 gives 155 nowhere to go. All three rules come down to one idea: every data value must fit into exactly one interval, not two and not none. Check the width by counting the values, since 150 to 154 covers 150, 151, 152, 153 and 154, which is five.

One more column is often useful: cumulative frequency, a running total that answers 'how many so far?'. Start with the first frequency, 4. Then add the next, so 4 plus 9 is 13. Then 13 plus 11 is 24, and 24 plus 5 is 30. Each entry is the one above it plus this row's frequency. Now a question like 'how many students are under 165 centimetres?' is read straight off as 24, with no adding needed. And the column checks itself: the last cumulative value must equal the total number of data values, which here is 30 ✓. If it does not, there is an error somewhere in the table.