Another problem faced when collecting data is that some data may be missing. For example, in conducting a survey with ten questions, perhaps some of the people who take the survey don’t answer all ten questions. In Identifying Outliers and Missing Data we show how to identify missing data using a supplemental data analysis tool provided in the Real Statistics Resource Pack.

A simple approach for dealing with missing data is to throw out all the data for any sample missing one or more data elements. One problem with this approach is that the sample size will be reduced. This is particularly relevant when the reduced sample size is too small to obtain significant results in the analysis. In this case additional sample data elements may need to be collected. This problem is a bigger than might first be evident. E.g. if a questionnaire with 5 questions is randomly missing 10% of the data, then on average almost 60% of the sample will have at least one question missing.

Also it is often the case that the missing data is not randomly distributed. E.g. people filling out a long questionnaire may give up at some point and not answer any further questions, or they may be offended or embarrassed by a particular question and choose not to answer it. These are characteristics that might be quite relevant to the analysis.

In general there are the following types of remedies for missing data:

- Delete the samples with any missing data elements
- Impute the value of the missing data
- Remove a variable (e.g. a particular question in the case of a questionnaire or survey) which has a high incidence of missing data, especially if there are other variables (e.g. questions) which measure similar aspects of the characteristics being studied.

**Deleting Missing Data**

Of particular importance is the randomness of the missing data. E.g. suppose a lot of people didn’t answer question 5 but everyone answered question 7. If the frequency of the responses to question 7 changes significantly when samples which are missing responses to question 5 are dropped, then the missing data is not random, and so dropping samples can bias the results of the analysis. In this case either another remedy should be employed or the analysis should be run twice: once with samples with missing data retained (e.g. by adding a “no response” for missing data) and once with these samples dropped.

Missing data can be removed by using the following supplemental Excel functions found in the Real Statistics Resource Pack.

**Real Statistics Functions**:

**DELBLANK**(R1, *s*) – fills the highlighted range with the data in range R1 (by columns) omitting any empty cells

**DELNonNum**(R1, *s*) – fills the highlighted range with the data in range R1 (by columns) omitting any non-numeric cells

**DELROWBLANK**(R1, *b, s*) – fills the highlighted range with the data in range R1 omitting any row which has one or more empty cells; if *b* is TRUE then the first row of R1 (presumably containing column headings) is always copied (even if it contains an empty cell); this argument is optional and defaults to *b* = FALSE.

**DELROWNonNum**(R1, *b, s*) – fills the highlighted range with the data in range R1 omitting any row which has one or more non-numeric cells; if *b* is TRUE then the first row of R1 (presumably containing column headings) is always is always copied (even if it contains a non-numeric cell); this argument is optional and defaults to *b* = FALSE.

The string *s* is used as a filler in case the output range has more cells/rows than needed. This argument is optional and defaults to the error value #N/A. See Data Conversion and Reformatting for an example of the use of these functions.

In addition there is the supplemental function **CountFullRows**(R1, *b*) where *b* = TRUE (default) or FALSE and

**CountFullRows**(R1, TRUE) = the number of rows in range R1 which don’t have any empty cells

**CountFullRows**(R1, FALSE) = the number of rows in range R1 which don’t have any non-numeric cells

There is also the related supplemental function **CountPairs**(R1, R2, *b*) where *b* = TRUE (default) or FALSE. Here we look at pairs of cells from R1 and R2: the *i*th cell in R1 is paired with the *i*th cell in R2

**CountPairs**(R1, R2, TRUE) = the number of pairs for which neither cell in the pair is empty

**CountPairs**(R1, R2, FALSE) = the number of pairs for which neither cell in the pair is empty or non-numeric

Note that in standard Excel the equivalent of **CountPairs**(R1, R2, TRUE) can be calculated by

=SUMPRODUCT(1-ISBLANK(R1),1-ISBLANK(R1))

**CountPairs**(R1, R2, FALSE) can be calculated by

=SUMPRODUCT(–ISNUMBER(R1),–ISNUMBER(R2))

To calculate the number of pair with equal numeric entries, we can use the formula

=SUMPRODUCT(–ISNUMBER(R1),–ISNUMBER(R2),–(R1=R2))

**Example 1**: Delete any missing data listwise (indicated by an empty cell) from the sample in A3:G22 in Figure 1.

**Figure 1 – Listwise deletion of missing data**

Since we want to delete any row which contains one or more empty cells (except the first row which contains column titles), we use the following array formula to produce the output in range I3:O22 of Figure 1: =DELROWBLANK(A3:G22,TRUE).

The cell M24 contains the formula =CountFullRows(A4:G22).

**Real Statistics Tools**: The Real Statistics Resource Pack supplies the **Reformatting a Data Range by Rows** data analysis tool which provides easier-to-use versions of the supplemental DELROWBLANK and DELROWNonNum functions described above.

We can also use the supplemental **Reformatting a Data Range** data analysis tool as substitutes for the DELBLANK and DELNonNum functions. We won’t demonstrate this tool here, but see Data Conversion and Reformatting for more information about how to use that tool.

**Example 2**: Repeat Example 1 using the **Reformatting a Data Range by Rows** data analysis tool.

To use this data analysis tool press **Ctrl-m** and choose the **Reformatting a Data Range by Rows** option. A dialog box will appear as in Figure 2. Fill in the dialog box as indicated and click on **OK**. The exact same output will appear as we saw previously (namely range I3:O22 of Figure 1).

**Figure 2 – Dialog box for Reformat Data Range by Rows**

The data analysis tool will output the same number of rows as in the input data range, but any extra rows would be filled in with the values #N/A. Since four rows had at least one empty cell, four rows are deleted from the output (those for Arkansas, Colorado, Idaho and Indiana) and so the last four rows of the output need to be filled with #N/A.

Actually all the cells in the output range I3:O22 will contain the array formula =DELROWBLANK(A3:G22,TRUE) and so if we change the value of cell B15 to say 10.2, the row for Idaho would now automatically appear in the output and there would be one less row with values #N/A.

If we had entered an asterisk in the **Filler** field of Figure 2, then the output would be the same as we saw in Figure 1 except that this time all the cells in range I19:O22 would contain an asterisk instead of #N/A.

If we had entered the number 0 in the **Filler** field then all the cells in the output range would contain the array formula =DELROWBLANK(A3:G22,True,””), although the values of all the cells in the range I19:O22 would be empty. As before if we change the contents of cell B15 to 10.2, then the row for Idaho would appear in the output and only three rows with empty cells would appear. All the cells in the output range would still have the same array formula, namely =DELROWBLANK(A3:G22,TRUE,””).

If we had checked the **Freeze output range size** element then the data analysis tool would determine that four rows have missing data and so it would output a range with four fewer rows, namely the range I3:O18. Although the output would be displayed exactly as in the case described in the previous paragraph, this time only the range I3:O18 would contain the formula =DELROWBLANK(A3:G22,TRUE). This time if cell B15 is changed to 10.2, then Idaho would be added to the output range, but since the output range only goes down to row 18, the last input row (that for Maine) would not be displayed, which is probably not what we want.

In conclusion, the **Freeze output range size** option makes the output cleaner (since all the rows contain data), but should not be used if there is the possibility that some missing data may be added later.

**Imputing the values for missing data**

Some techniques for imputing values for missing data include:

- Substituting the missing data with another observation which is considered similar, either taken from another sample or from a previous study
- Using the mean of all the non-missing data elements for that variable. This might be acceptable in cases with a small number of missing data elements, but otherwise it can distort the distribution of the data (e.g. by reducing the variance) or by lowering the observed correlations (see Basic Concepts of Correlation).
- Using regression techniques. In this approach regression (as described in Regression and Multiple Regression) is used to predict the value of the missing data element based on the relationship between that variable and other variables. This approach reinforces existing relationships and so makes it more likely that the analysis will characterize the sample and not the general population.

I am running a regression analysis in which I want to predict housing prices based upon variables in previously sold homes (number of bedrooms, bathrooms, year built, amenities, etc) In many cases the historical data is missing information on one or more variables. If I delete the row of data in which a missing value occurs, I’m going to lost a LOT of cases. Will Excel not run regression if a cell is left blank? If that is the case, is there a work around?

Kem,

Excel will not run regression with missing data. I describe various techniques for dealing with missing data, especially for regression on the following webpage:

Handling Missing Data

Charles

How can I replace missing data for questions that didn’t appear to respondents.

I have logic skip questions in my survey, so if for example Q 2 has a scale from 1-6 and the respondent gave me an answer on that scale from 1-3. Q 3 shouldn’t be displayed to the respondent. I have 40 cases missing in my data for that question only because the Q3 wasn’t displayed to the respondents

Shaimaa,

How you deal with missing data depends on what you plan to do next. Assuming that there was no missing data, what sort of analyses are you planning to do?

Charles

I am very elementary at statistics. I am working with total population numbers for each year from 2000 to 2010. I am missing the total population for the year 2003. The total population for year 2002 is 1012 and for year 2004 is 1146. What is the best method to to replace the missing population number for year 2003? What is the name of this best method? How valid will this method be?

Roy,

This is a problem in forecasting time series with missing data. I am now working on adding tutorials about time series to the website and data analysis tools and functions to the Real Statistics Resource Pack.

There are a number of papers that deal with the subject that you are asking about. Here is one that may be helpful to you.

http://www.mff.cuni.cz/veda/konference/wds/proc/pdf08/WDS08_111_m5_Hanzak.pdf

Charles

I am trying to find an appropriate way to handle missing data due to the death of the subject when that death is influenced by treatement.

i.e. Treatment A is supposed to alleviate the symptoms of a disease and to delay death in animal subjects relative to treatment B.

Regular measurements of weight and clinical score are taken from subjects. The study is continued till many of the subjects have become moribund (when they are euthenized)

Analysis of date of death is straightforward, but how can I analyze or impute data for body weight and clinical score when the worst affected animals are gone?

Ambrose,

The typical approaches to imputing values to missing data are based on the assumption that such data are missing at random (with various definitions of what this means). Since this is not your case, you can’t use any of these techniques.

I would need some more information about what sorts of data you are referring to. If it is weight, then perhaps you could use the weight just before or at death. If it is heart rate, perhaps you could use the heart rate just before death or the value zero. In other words, the values assigned to “missing” data probably don’t rely on mathematical techniques, but approaches based on your domain knowledge.

Charles

I would like to compute the reduction percentage only of certain data (A and B) and wherein I have values as eg:

1. A=1 and B=2 it resulted to 200% decreased, in which it has corresponding points of 75 for every 75-100% decreased;

2. A=2 and B=1 it resulted to 50% decreased with same corresponding pts of 50 for every 50-74% increased;

3. A=0 and B=0, or maintained values like A=5 and B=5; which has no reduction nor increased. Wherein the corresponding pts if there is reduction from 0-25% is 20;

4. A=1 and B=0 which resulted to an error;

5. A=0 and B=1 with 0 result which conflicted to number 3 example (A=0 and B=0);

And also with blank/empty cell.

Sorry Bryan, but I don’t understand your question. Are you looking for a single value which gives the overall percentage of reduction?

Charles

Hi Sir Charles,

I have a degree in statistics but it’s not enough for me to face common challenges particularly in analyzing survey results which commonly use only frequency and percentage.

However, I have a question. I know it is very simple for you but I am just not sure what is the right way of doing so. This is just frequency and percentage, not really advance stat analysis etc. But how should I specifically handle this case: we ask the sample if they have business, if yes it will proceed to question like “did you already already accessed loan?”. Say having business is 256, but next question shows that accessed loan=100 and did not accessed=150 which means there is 6 is missing. How this case should be dealt particularly on computing the percent frequency? should it be accessed loan: 100/250 or 100/256??

It really depends on how you will use the information subsequently, but based on what I have heard so far, I would say 100/250.

Charles

How do I replace missing values with the mean. My data is from B2:B73 and the missing data is within that range

Amy,

One way to do it is to create a copy of the data in column D with the missing data filled in. Assuming missing data is an empty cell, this can be accomplished by

1. Highlight the range D2:D73

2. Insert the formula =IF(B2<>“”,B2,AVERAGE(B$2:B$73)) in cell D2

3. Press Ctrl-D

Charles

This function is not found in normal excel function for Microsoft Office 2007 and 2010. But I do not know if it is contained in other advanced Microsoft Office. This is very useful especially with very huge data.

Not sure which function you are referring to, but the functions described on this part of the website are part of the Real Statistics Resource Pack, and are not contained standard Excel.

Charles

I’m calculating the min, median, max, mean, standard deviation of a set of data with missing values in it using excel. How do I replace those missing data by mean or by median in excel? And when is replacement by median suitable?

First note that when calculating the min, median, max, mean and standard deviation Excel ignores any missing data.

Suppose your data is in range A1:D10 and the mean or median is in cell A12 and missing data is indicated by an empty cell. Now place the formula =IF(A1=””,A12,A1) in cell F1 and then highlight the range F1:I10 and press Ctrl-R and Ctrl-D. Range F1:I10 will now contain the desired values.

As explained on the website, replacing missing data by the mean (or median) will reduce the variance of the data and so is undesirable. About the only time you should even consider doing this is if only a very small percentage of the data is missing.

Charles

I am trying to calculate individuals’ variances on multiple measures with missing data coded as -99. I would like my formula to calculate the variance for these measures while ignoring/skipping/omitting the -99 values.

I do not want to change all of the -99 values to 0 or blank because it is a an indicator of how many questions they saw but chose to skip in a survey with lots of branching. If I wanted to get rid of them I would just use ctrl+f and replace them with “”. Any suggestions other than using SPSS or other statistical softwares?

Currently my formula is: =IF(ISERROR(VAR(DJ4:EH4)),””,VAR(DJ4:EH4))

But, it could be simplified to =VAR(DJ4:EH4) to figure out how to do this first.

I would like it to be something like: calculate the var(DJ4:EH4) except for cells = -99.

I tried =Var(DJ4:EH4, <"-99"), but that didn't work.

Tiffany,

The the data is in range R1 then the following formula should work. Keep in mind that this is an array formula and so you need to press Ctrl-Shft-Enter.

=VAR(IF(R1=-99,””,R1))

Charles