Frequency Tables

Often data is presented in the form of a frequency table. For example, the data in range A4:A11 of Figure 1 can be expressed by the frequency table in range C4:D7.

Frequency table Excel

Figure 1 – Frequency Table

The table in Figure 1 shows that the data element 2 occurs 4 times, the element 4 occurs 2 times and the element 3 and 5 occur 1 time.

When data is provided in the form of a frequency table, the calculation of the mean and standard deviation cannot be performed directly using the usual AVERAGE and STDEV Excel functions. In fact for sample data  {x1, …, xm} with corresponding frequency counts of f1, …, fm respectively and sample size n = f1 + f2 + … + fm, then the sample mean is (see Measures of Central Tendency):

Mean frequency table

This can be calculated in Excel as

=SUMPRODUCT(R1, R2) / SUM(R2)

where R1 is a range containing the data elements {x1, …, xm}  and R2 is a range containing  {f1, …, fm}.

In a similar way, the sample variance can be calculated as

image7106

which can be calculated in Excel as

=SUMPRODUCT((R1-R3)^2, R2)/(SUM(R2)-1)

where R1 and R2 are as above and R3 contains the sample mean (as described above). Using Property 1 of Measures of Variability, we also have the following alternative approach to calculating the variance:

image101x

which can be calculated in Excel by the formula

=(SUMPRODUCT(R1^2,R2)-SUM(R2)*R3^2)/(SUM(R2)-1)

Using these formulas we can calculate the mean and variance of sample data expressed in the form of a frequency table. We demonstrate this in the following example.

Example 1: Calculate the mean and variance of the sample data from the frequency table in Figure 1.

Frequency table mean variance

Figure 2 – Calculation of mean and standard deviation from frequency table

The required calculation is displayed in Figure 2. Here cell F11 contains the formula =D11/E11, which calculates the mean, and cell G14 contains the formula =(D14-E14*F14)/(E14-1), which calculates the variance. The results are the same as calculating the mean and variance by applying Excel’s AVERAGE and VAR functions to the data set {2, 2, 2, 2, 3, 4, 4, 5}.

Note too that a frequency table is closely linked to a frequency function, as defined in Definition 1 of Discrete Distributions. E.g. since there are 8 elements in the data set in Figure 2, we see that the frequency function for the random variable x is as in Figure 3 where each frequency value is divided by 8:

Frequency function via table

Figure 3 – Frequency function corresponding to frequency table

Often frequency tables are used with a range of data values, i.e. with intervals for the x values. In this case the midpoint of each interval is assigned the value xi.

Example 2: Calculate the mean and variance for the data in the frequency table in Figure 4.

Frequency table with intervals

Figure 4 – Calculations for a frequency table with intervals

The first interval in Figure 4 is 0 < x ≤ 4, the second 4 < x ≤ 10, etc. The calculation of the mean and variance is as in Figure 2, except that now the midpoints are used as the x values.

Observation: Sometimes the first and/or last interval is unbounded: e.g. if the last interval in Figure 4 is replaced by “over 20”. In this case it isn’t possible to establish a midpoint, and so all you can do is make your best estimate of a suitable representative value for that interval.

Excel Function: When you have a lot of data, it is convenient to put the data in bins, usually of equal size, and then create a graph of the number of data elements in each bin. Excel provides the FREQUENCY(R1, R2) array function for doing this, where R1 = the input array and R2 = the bin array.

To use the FREQUENCY array function, enter the data into the worksheet and then enter a bin array. The bin array defines the intervals that make up the bins. E.g., if the bin array = 10, 20, 30, then there are 4 bins, namely data with values x ≤ 10, data with values x where 10 < x ≤ 20, data with values x where 20 < x ≤ 30, and finally data with values x > 30. The FREQUENCY function simply returns an array consisting of the number of data elements in each of the bins.

Example 3: Create a frequency table for the 22 data elements in the range A4:B14 of Figure 5 based on the bin array D4:D7 (the text “over 20” in cell D8 is not part of the bin array).

Frequency function Excel

Figure 5 – Example of the FREQUENCY function

To produce the output, highlight the range E4:E8 (i.e. a column range with one more cell than the number of bins) and enter the formula

=FREQUENCY(A4:B11,D4:D7)

Since this is an array formula, you must press Ctrl-Shft-Enter. Excel now inserts frequency values in the highlighted range E4:E8. Here E4 contains the number of data elements in the input range with value in the first bin (i.e. data elements whose value is ≤ 20). Similarly, E5 contains the number of data elements in the input range with value in the second bin (i.e. data elements whose value is > 20 and ≤ 40). The final output cell (E8) contains the number of data elements in the input range with value > the value of the final bin (i.e. > 80 for this example).

Observation: As described in Discrete Probability Distributions, the Real Statistics Resource Pack provides the FREQTABLE function. This function can also be used to create a frequency table with bins where the bins are equally spaced.

Real Statistics Function: The Real Statistics Resource Pack supplies the following supplemental array function to create a frequency table

FREQTABLE(R1, bsizebmax) = an array function which produces the frequency table for the data in range R1, assuming equally sized bins of size bsize where bmax is the maximum bin size value

To use the function you must highlight an array with 3 columns and at least k rows where k = (bmax – MIN(R1)) / bsize + 1. You can highlight more rows than you need; any extra rows will take value #N/A.

The bmax argument is optional. If omitted then it defaults to bmax = MAX(R1). When bmax is not omitted then you should make sure that bmax ≥ MAX(R1): otherwise some data will be lost.

The bsize argument is also optional. If omitted then instead of creating a frequency table as described above, a table with a bin for each value in R1 is used. The bmax argument is ignored. This produces a table similar to that described in Example 3 of Discrete Distributions.

Example 4: Create a frequency table for the 22 data elements in the range A4:B14 of Figure 5 based on bins of size 15.

The desired frequency table can be produced using the array formula

=FREQTABLE(A4:B14,15)

as shown in range M4:O11 of Figure 6.

FREQTABLE function

Figure 6 – FREQTABLE function with bin size 15

The headings are not outputted by the function but have been added manually. Note that two extra rows have been highlighted and so they are filled with #N/A.

Observation: You can also use the Frequency Table data analysis tool for creating frequency tables. See Histograms for an example of how to use these tools. Also see Frequency Table Conversion for how to calculate the descriptive statistics (see Descriptive Statistics) for the data described by a frequency table.

11 Responses to Frequency Tables

  1. Alexandre S Marcal says:

    Hello Prof. Charles;

    Thanks for all the tips.

    Question: When using the native Frequency function (and when using routines that rely on Frequency function, such as creating Histograms), Excel rounds up every number that exceeds 50% of the range of the bin. For example, if I input bins 10, 20, 30…N, any numbers in my data matrix ranging from [10-15 will fall under bin 10, while numbers ranging from [15-25 will fall under bin 20, and so on. This is really a nuisance when trying to select the bins. Is there any way to stop Excel from rounding up the numbers before assorting them to the bin (i.e. force Excel to assort the number to the specified bin)? Thanks in advance.

    • Charles says:

      Alexandre,
      This is not how the FREQUENCY function works. If your bins are 10,20,30 and your data are 8, 12, 17, 20, 22, 28, then the frequency values for 10, 20, 30 are respectively 1, 3, 2. In particular, the frequency value for 20 (from 0,20,30) is the count of all data values larger than 10 and less than or equal to 20.
      Charles

  2. Frank Medrisch says:

    hEY,

    How do you do the same thing when you have X and f(x)…it is not explained…

  3. Theo Kirkland says:

    Hi -Thanks very much for this statistics tool. I’m trying to make a frequency table on EXCEL for Mac released in late 2015. I got data from one column, but I can’t get the formula to work again. Do you know of a helpful resource for Excel on the Mac?

  4. Wytek says:

    Hi Charles,
    Before anything, I must congratulate you on your website. One of the best learning sites for statistics.

    But with respect to the Frequency Tables page I recommend that you explain the computation of the mean and variance in terms of the first and second moments, e.g. E[X] and [EX^2]. This is what we have learned in college.

    Mean = E[X] and var(X) = E[X^2] – (E[X])^2.

    You sort of compute the first and second moments in figure 2 but it should be done on the probability table in figure 3. Any thoughts?

    • Charles says:

      Wytek,
      I have revised the referenced webpage to make things clearer. I have referenced Property 1 on another webpage which captures the moments idea.
      For the probability table in Figure 3, I wanted to emphasize the idea of dividing by the sample size.
      Charles

  5. Bob Larson says:

    This is terrific! Thank you for sharing.

  6. Douglas Clark says:

    Question: What if I am calculating frequencies of nominal (categorical) data? Eg. frequencies of face-to-face contacts, phone call contacts, emails, and letters. I am preparing an electronic survey that contains a lot of these types of choices. In addition, there are open-ended questions whose responses will be put into categorical bins and displayed as frequencies. I ultimate question is this, “How do I determine a sample size for a survey with these kinds of questions? The population is 800. If you can give me any guidance or send me a reference, I would be most grateful. Thanks, Doug Clark

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