Frequency Tables
Imagine you ask everyone in your class what their favourite fruit is. Twenty-eight answers come
flying at you — apple, banana, banana, grapes, apple, apple… — far too fast to remember. You need
a tidy way to record how often each answer comes up. That tidy way is a
frequency table.
A frequency table records how many times each value or category occurs in a set
of data. It has a row for every different answer, and a column that says how many of that answer
there are. That count is called the frequency — the word just means
"how often".
Tally marks: counting in fives
While the answers are still flying in, you cannot stop to recount the whole list each time. So you
keep a running count with tally marks — one little stroke for each answer. To stop
a long row of strokes becoming a blur, we bundle them into groups of five:
- The first four answers are four upright strokes: ||||.
-
The fifth answer is not a fifth upright stroke — it is a diagonal line
drawn straight across the other four, like a bar across a gate. That finished
"gate" of five is easy to spot at a glance.
-
Then you start a new gate for the next five, and so on. To total a tally you count the gates in
fives and add the leftover strokes: two gates and three strokes is
5 + 5 + 3 = 13.
Counting in fives is what makes a tally so fast: instead of counting twenty separate marks you
count four neat gates. Once the data has all arrived, you turn each tally into a number — its
frequency — and write it in the last column.
A worked example
Suppose a shop notes the shoe size of every customer one morning. As each customer comes in, the
assistant adds a tally stroke next to their size. At the end of the morning the tallies are turned
into frequencies:
| Value (shoe size) | Tally | Frequency |
|---|
| 4 | ||| | 3 |
| 5 | ̶|||| (one gate) | 5 |
| 6 | ̶|||| ̶|||| ||| (two gates + 3) | 8 |
| 7 | |||| (four) | 4 |
| Total | | 20 |
Notice the frequencies 3 + 5 + 8 + 4 = 20 add up to the total number of
customers. Every customer is counted exactly once, so the frequencies must sum to
the size of the whole data set. That sum is the total frequency, and it is a handy
check: if your frequencies do not add up to the number of items you started with, a tally has gone
wrong somewhere.
Size 6 has the biggest frequency, so size 6 is
the mode — the most common value in the data.
See it: a tally chart
Here is a class fruit survey laid out as a tally chart. Each fruit has a row; the tally strokes are
bundled into gates of five; and the last column turns each tally into its
frequency. Read the gates in fives, add the leftover strokes, and check it matches
the frequency. The bottom row adds the frequencies to give the total number of children surveyed.
Press Refresh for a brand-new survey.
Suppose you ask ten friends their favourite fruit and they say: apple, banana, apple, grapes,
apple, banana, grapes, apple, banana, grapes. Go through the list once, adding a stroke each time:
apple ̶|||| = 4
banana ||| = 3
grapes ||| = 3
The frequencies are 4 + 3 + 3 = 10 — exactly the ten friends you asked.
Apple has the highest frequency, so apple is the mode: the favourite of the group.
Try counting twenty separate strokes in a row — it is slow and easy to lose your place. Now picture
four finished gates instead: you read "five, ten, fifteen, twenty" in a heartbeat. The gate is a
tiny invention that turns a hard counting job into easy skip-counting. That is the whole reason the
fifth stroke goes across the gate instead of beside it.
The two classic tally-and-frequency traps:
-
The fifth tally is the line drawn across the gate, not a fifth
upright stroke beside it. A finished gate is always five, so total a tally by
counting in fives: gate, gate, gate… then add the leftovers.
-
The frequencies must add up to the number of items you collected. If you
surveyed 20 children but your frequencies sum to 19 or 21, you have miscounted a tally — go back
and check.
See it explained