The Mean from a Frequency Table

It is easy to calculate the Mean:

Add up all the numbers,
then divide by how many numbers there are.

Example: What is the Mean of these numbers?

6, 11, 7

  • Add the numbers: 6 + 11 + 7 = 24
  • Divide by how many numbers (there are 3 numbers): 24 ÷ 3 = 8

The Mean is 8

But sometimes we don't have a simple list of numbers, it might be a frequency table like this (the "frequency" says how often they occur):

Score Frequency
1 2
2 5
3 4
4 2
5 1

(it says that score 1 occurred 2 times, score 2 occurred 5 times, etc)

We could list all the numbers like this:

Mean = 1+1 + 2+2+2+2+2 + 3+3+3+3 + 4+4 + 5(how many numbers)

But instead of lots of adds (like 3+3+3+3) it is easier to use multiplication:

Mean = 1 + 2 + 3 + 4 + 5 (how many numbers)

And instead of counting how many numbers, we can add up the frequencies:

Mean = 1 + 2 + 3 + 4 + 5 2 + 5 + 4 + 2 + 1

So now we calculate:

Mean = 2 + 10 + 12 + 8 + 5 14  
=  3714  =  2.64...

And that is how to calculate the mean from a frequency table!

Here is another example:

Example: Parking Spaces per House in Hampton Street

Isabella went up and down the street to find out how many parking spaces each house has. Here are her results:

Parking
Spaces
Frequency
1 15
2 27
3 8
4 5

What is the mean number of Parking Spaces?

Answer:

Mean = 15×1 + 27×2 + 3 + 415 + 27 + 8 + 5
15 + 54 + 24 + 2055
2.05...

The Mean is 2.05 (to 2 decimal places)

(much easier than adding all numbers separately!)

Notation

Now you know how to do it, let's do that last example again, but using formulas.

Σ
This symbol (called Sigma) means "sum up"
(read more at Sigma Notation)
Σf
And this means "sum up all the frequencies"
(where f is frequency)

And we can use it like this:

sum f

Likewise we can add up "frequency times score" this way:

sum fx
(where f is frequency and x is the matching score)

And the formula for calculating the mean from a frequency table is:

mean of a frequency table

The x with the bar on top says "the mean of x"

So now we are ready to do our example above, but with correct notation.

Example: Calculate the Mean of this Frequency Table

x f
1 15
2 27
3 8
4 5

And here it is:

Mean of x =
ΣfxΣf
=
15×1 + 27×2 + 8×3 + 5×415+27+8+5
=
2.05...

There you go! You can use sigma notation.

Calculate in the Table

It is often better to do the calculations in the table.

Example: (continued)

From the previous example, calculate f  × x in the right-hand column and then do totals:

x f fx
1 15 15
2 27 54
3 8 24
4 5 20
TOTALS: 55 113

And the Mean is then easy:

Mean of x = 11355 = 2.05...

 

694, 695, 1464, 1465, 1466, 1467, 3060, 3061, 3062, 3063