We consider a random variable *x* and a data set *S = *{*x*_{1}*, x*_{2}*, …, x _{n}*} of size

*n*which contains possible values of

*x*. The data set can represent either the population being studied or a sample drawn from the population. We can also view the data as defining a distribution, as described in Discrete Probability Distributions.

We seek a single measure (i.e. a statistic) which somehow represents the center of the entire data set *S*. The commonly used measures of central tendency are the **mean**, **median** and **mode**. Besides the normally studied mean (also called the **arithmetic **mean) we also consider two other types of mean: the **geometric **mean and the **harmonic** mean.

**Excel Functions**: If R is an Excel range which contains the data elements in *S* then the Excel formula which calculates each of these statistics is shown in Figure 1.

Statistic |
Excel 2007 |
Excel 2010/2013/2016 |

Arithmetic Mean | AVERAGE(R) | AVERAGE(R) |

Median | MEDIAN(R) | MEDIAN(R) |

Mode | MODE(R) | MODE.SNGL(R), MODE.MULT(R) |

Geometric Mean | GEOMEAN(R) | GEOMEAN(R) |

Harmonic Mean | HARMEAN(R) | HARMEAN(R) |

**Figure 1 – Measures of central tendency**

**Observation**: All these functions ignore any empty or non-numeric cells.

While formulas such as AVERAGE(R1) (as well as VAR(R), STDEV(R), etc. described on other webpages) ignore any empty or non-numeric cells, they return an error value if R contains an error value such as #NUM or #DIV/0!. This limitation can often be overcome by using the following approach:

=AVERAGE(IF(ISERROR(R),””,R))

This array formula returns the mean of all the cells in R1 ignoring any cells that contain an error value. Since this is an array formula, you must press **Ctrl-Shft-Enter**. An alternative approach is to use the following function.

**Real Statistics Function**: The Real Statistics Resource Pack provides the following array function:

**DELErr**(R) = the array of the same size and shape as R consisting of all the elements in R where any cells with an error value are replaced by a blank (i.e. an empty cell).

E.g. to find the average of a range R which may contain error cells, you can use the formula

=AVERAGE(DELErr(R))

**Real Statistics Data Analysis Tool**: The **Remove error cells** option of the **Reformatting a Data Range **data analysis tool described in Reformatting Tools makes a copy of the inputted range where all cells that contain error values are replaced by empty cells.

To use this capability, press **Ctrl-m** and double click on **Reformatting a Data Range**. When the dialog box shown in Figure 2 of Reformatting Tools, fill in the **Input Range**, choose the **Remove error cells **option and leave the** # of Rows **and** # of Columns **fields blank. The output will have the same size and shape as the input range.

**Mean**

We begin with the most commonly used measure of central tendency, the mean.

**Definition 1**: The **mean** (also called the **arithmetic mean**) of the data set *S* is defined by

**Excel Function**: The mean is calculated in Excel using the function **AVERAGE**.

**Example 1**: The mean of *S* = {5, 2, -1, 3, 7, 5, 0, 2} is (2 + 5 – 1 + 3 + 7 + 5 + 0 + 2) / 8 = 2.875. We achieve the same result by using the formula =AVERAGE(C3:C10) in Figure 2.

**Figure 2 – Excel examples of central tendency**

**Observation**: When the data set *S* is a population the Greek letter µ is used for the mean. When *S* is a sample, then the symbol *x̄* is used.

**Observation**: When data is expressed in the form of frequency tables then the following property is useful.

**Property 1**: If *x̄* is the mean of sample {*x*_{1}*, x*_{2}*, …, x _{m}*} and

*ȳ*is the mean of sample {y

_{1}, y

_{2}, …, y

*} then the mean of the combined sample is*

_{n}Similarly, if* µ _{x}* is the mean of population {

*x*

_{1}

*, x*

_{2}

*, …, x*} and

_{m}*µ*

_{y}is the mean of population {y

_{1}, y

_{2}, …, y

*} then the mean of the combined population is*

_{n}**Real Statistics Functions**: The Real Statistics Resource Pack furnishes the following array functions:

**COUNTCOL**(R1) = a row range which contains the number of numeric elements in each of the columns in R1

**SUMCOL**(R1) = a row range which contains the sums of each of the columns in R1

**MEANCOL**(R1) = a row range which contains the means of each of the columns in R1

**COUNTROW**(R1) = a column range which contains the number of numeric elements in each of the rows in R1

**SUMROW**(R1) = a column range which contains the sums of each of the rows in R1

**MEANROW**(R1) = a column range which contains the means of each of the rows in R1

**Example 2**: Use the COUNTCOL and MEANCOL functions to calculate the number of cells in each of the three columns in the range L4:N11 of Figure 3 as well as their means.

**Figure 3 – Count, Sum and Mean by Column**

The array formula =COUNTCOL(L4:N11) produces the first result (in range L13:N13), while the formula =MEANCOL(L4:N11) produces the second result (in range L14:N14) and the formula =SUMCOL(L4:N11) produces the third result (in range L15:N15).

Remember that after entering any of these formulas you must press **Ctrl-Shft-Enter**.

**Median**

**Definition 2**: The **median** of the data set *S* is the middle value in *S*. If you arrange the data in increasing order the middle value is the median. When *S* has an even number of elements there are two such values; the average of these two values is the median.

**Excel Function**: The median is calculated in Excel using the function **MEDIAN**.

**Example 3**: The median of *S* = {5, 2, -1, 3, 7, 5, 0} is 3 since 3 is the middle value (i.e the 4th of 7 values) in -1, 0, 2, 3, 5, 5, 7. We achieve the same result by using the formula =MEDIAN(B3:B10) in Figure 2.

Note that each of the functions in Figure 2 ignores any non-numeric values, including blanks. Thus the value obtained for =MEDIAN(B3:B10) is the same as that for =MEDIAN(B3:B9).

The median of *S* = {5, 2, -1, 3, 7, 5, 0, 2} is 2.5 since 2.5 is the average of the two middle value 2 and 3 of -1, 0, 2, 2, 3, 5, 5, 7. This is the same result as =MEDIAN(C3:C10) in Figure 2.

**Mode**

**Definition 3**: The **mode** of the data set *S* is the value of the data element that occurs most often.

**Example 4**: The mode of *S* = {5, 2, -1, 3, 7, 5, 0} is 5 since 5 occurs twice, more than any other data element. This is the result we obtain from the formula =MODE(B3:B10) in Figure 2. When there is only one mode, as in this example, we say that *S* is **unimodal**.

If *S* = {5, 2, -1, 3, 7, 5, 0, 2}, the mode of *S* consists of both 2 and 5 since they each occur twice, more than any other data element. When there are two modes, as in this case, we say that *S* is **bimodal**.

**Excel Function**: The mode is calculated in Excel by the formula **MODE**. If range R contains unimodal data then MODE(R) returns this unique mode. For the first data set in Example 3 this is 5. When R contains data with more than one mode, MODE(R) returns the first of these modes. For the second data set in Example 4 this is 5 (since 5 occurs before 2, the other mode, in the data set). Thus MODE(C3:C10) = 5.

As remarked above, if there is more than one mode, MODE returns only the first, although if all the values occur only once then MODE returns an error value. This is the case for *S* = {5, 2, -1, 3, 7, 4, 0, 6}. Thus MODE(D3:D10) = #N/A.

Excel 2010/2013 provide an array function **MODE.MULT**, which is useful for multimodal data by returning a vertical list of modes. When we highlight C19:C20 and enter the array formula =MODE.MULT(C3: C10) and then press **Ctrl-Alt-Enter**, we see that both modes are displayed.

Excel 2010/2013 also provide the function **MODE.SNGL** which is equivalent to MODE.

**Geometric Mean**

**Definition 4**: The **geometric mean** of the data set *S* is calculated by

This statistic is commonly used to provide a measure of average rate of growth as described in Example 5.

**Example 5**: Suppose the sales of a certain product grow 5% in the first two years and 10% in the next two years, what is the average rate of growth over the 4 years?

If sales in year 1 are $1 then sales at the end of the 4 years are (1 + .05)(1 + .05)(1 + .1)(1 + .1) = 1.334. The annual growth rate *r* is that amount such that (1+*r*)^{4} = 1.334. Thus *r* = 1.334^{1/4} – 1 = .0747.

The same annual growth rate of 7.47% can be obtained in Excel using the formula GEOMEAN(H7:H10) – 1 = .0747.

**Harmonic Mean**

**Definition 5**: The **harmonic mean** of the data set *S* is calculated by the formula

The harmonic mean can be used to calculate an average speed, as described in Example 6.

**Example 6**: If you go to your destination at 50 mph and return at 70 mph, what is your average rate of speed?

Assuming the distance to your destination is *d*, the time it takes to reach your destination is *d*/50 hours and the time it takes to return is *d*/70, for a total of *d*/50 + *d*/70 hours. Since the distance for the whole trip is 2*d*, your average speed for the whole trip is

This is equivalent to the harmonic mean of 50 and 70, and so can be calculated in Excel as HARMEAN(50,70), which is HARMEAN(G7:G8) from Figure 2.

Which measure of central tendency is considered for finding the average rates?

Siddhi,

It depends on what you mean by average rates. As you can see from the referenced webpage, the harmonic mean is often used for this purpose.

Charles

Hi Charles. How are you? I just would like to ask on how you do you get the weighted mean in excel? I have a data which involve 10 raters and their responses on the 19 items survey questionnaire using a 5-point likert scale. How do I get the weighted mean and likewise how do I get the inter-rater reliability index using Fleiss’s Kappa reliability coefficient? Thank you and God bless!

Erdz,

If your data is in range A1:A10 and the weights you use are in range B1:B10, then the weighted average can be calculated by the formula =SUMPRODUCT(A1:A10,B1:B10)/SUM(B1:B10)

Thanks for sending me your data. I can see that it is not formatted correctly for using the Real Statistics data analysis tool. Please make sure that you use the format shown in Example 1 of http://www.real-statistics.com/reliability/fleiss-kappa/

Charles

Hi, Charles,

Thanks for creating the web site and tools!

Could you explain a bit more about the “general rules” about when to use the harmonic and geometric means? I understand the examples you gave here, but I am curious about why they are called these ways. Why is geometric mean “geometric” and when it should be used? Why harmonic mean is “harmonic” and what problem does it solve?

Thanks!

Xiaobin,

See the following webpages:

https://en.wikipedia.org/wiki/Harmonic_mean

https://en.wikipedia.org/wiki/Geometric_mean

Charles

what is square root of 10.14?

Use the formula =SQRT(10.14)

Charles

Minor typo:

I think a final bracket is missing in “=AVERAGE(IF(ISERROR(R),””,R)” to make the formula work.

Jan,

Yes, you are correct. Thanks for catching this typo.

I really appreciate you help in improving the website.

Charles

Dear Sir,

I’m afraid that right equation for Example 5 (Geometric Mean) is:

r = 1.334^(1/4) = .0747, but not r = .334^(1/4) =0.0747.

That is, of course, only formal and negligible remark. Many thanks, indeed, for your great work.

Best Regards,

Petr Mrak.

Thank you Petr, I have now corrected the mistake that you have identified.

I appreciate your help in making the website more accurate.

Charles

I have a question about the geometric mean example.

If the growth rates from each year are .05, .05, 0.1 and 0.1 the GEOMEAN of these numbers is 0.070711….like Barb mentioned.

But when you list them like Charles does in the example (1.05, 1.05, 1.1, 1.1) and do GEOMEAN() – 1 then you get 0.0747.

To me the first example strikes as the one that’d be most frequently encountered in the real world.

So I’m not sure why the latter is assumed to be correct but for answer from the former is not.

Any clarification would be appreciated.

Jonathan,

I thought that Barb latter responded that she understood how I calculated my answer. How did you calculate 0.070711?

Charles

Charles,

0.070711 is calculated as =GEO.MEAN(.05 ; .05 ; 0.1 ; 0.1), i.e. the geometric mean of the %. This result is not meaningful, in that sense that you can’t do anything with it.

If on the other hand you calculate =GEO.MEAN(1.05 ; 1.05 ; 1.10 ; 1.10) -1 , you get .0747. This result is meaningful in that sense that it is the average, annual growth. If you calculate “start amount . (1+ .0747)^4” you get the amount as if sales went up .0747 (7,477%) each year during 4 years.

The reason for calculating with “1.05” and “1.10” instead of “0.05” and “0.10” is that

in general: A = a . (1+i)^n

(where A = end amount, a = start amount, i = annual growth (decimal) , n= number of years)

E.g. growth in 5 consecutive years: 5%; 6%; -3% (reduction); 10% and 2 %

Calculating the annual growth: =geo.mean (1.05 ; 1.06 ; 0.97 ; 1.10 ; 1.02) -1 = 0.039087. So the consecutive growths (and reduction) result in an average annual growth of 3.9%. The growth in five years equals a growth of 3.9% each year during 5 years.

If start amount = 25 (a), then end amount (A) after 5 years (n) = a . (1+i)^n = 25 . (1 + 0.039087)^5 = 30.2831.

Hopefully this helps Jonathan and others.

Jan

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it is of great help to me.

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please feel free to get my services

I’m not sure what is contained in H7:H10. I tried 0.05, 0.05, 0.1, 0.1. For GEOMEAN I got 0.070711.

I found the answer. It’s higher up on the page.