Frequency Tables

A frequency table consists of two columns, one column defines an interval and the other column is the number of values that the interval contains.

Before spreadsheets became widely available, using a frequency table to estimate summary statistics often required a lot less arithmetic than working with long columns of individual values.  A frequency table in graphic from is called a histogram.  A frequency table a good way of estimating the modal value of a set of numbers.

Example

The example dataset is based on observations of wind speed taken at noon.  As this is a real world dataset, a few readings are missing, thus there are only 337 values, rather than 365.

The data for this histogram is shown below.

Estimation of the Modal Value

The modal interval is the one which contains the greatest number of values.  In this case the modal interval is the one which contains values which are greater than 1 and less than or equal to 2.  As reasonable estimate of the mode is to take the mid-point of the modal interval which is 1.5.

Estimation of the Mean

 Two columns have been added, one for the mid-point of the interval and the other for the product of the mid-point and the frequency

Interval	Mid-point	Frequency	Frequency*Mid-Point
0	0	30	0
>0 and <=1	0.5	40	20.0
>1 and <=2	1.5	87	130.5
>2 and <=3	2.5	68	170.0
>3 and <=4	3.5	58	203.0
>4 and <=5	4.5	19	85.5
>5 and <=6	5.5	18	99.0
>6 and <=7	6.5	11	71.5
>7 and <=8	7.5	4	30.0
>8 and <=9	8.5	1	8.5
>9 and <=10	9.5	1	9.5
	Totals	337	827.5

The next step is to calculate the sums of Frequeny and Mid-point*Frequency

	Frequency	Frequency*Mid-point
Sums	337	827.5

The final stage is to calculate the mean, as shown below:

Mean=

sum(Mid-point*Frequency)/sum(Frequency)=

827.5/337=

2.46

The clip below shows how the calculation might be performed in Excel.

Comment

The method of estimating the mean described above, assumes that all the values in an interval are uniformly distributed, this may not be the case, especially in datasets such as the example which are left or right skewed.  Thus  the value of the mean obtained from a frequency table maybe slightly different from that obtained using the original data values.

Spreadsheets

The simplest way to compile a frequency table is to use the frequency function in Excel of Google Docs:

=frequency(data array,bin/class array)

The data array is the range of values to be analysed and the bin/class array is a column containing the upper bound of the interval, this is illustrated in the snip below: