Graphical Tests for Normality and Symmetry

Histogram

A histogram can be used to test whether data is normally distributed. This test simply consists of looking at the histogram and discerning whether it approximates the bell curve of a normal distribution.

Example 1: Determine whether the data in column B of Figure 1 are normally distributed using a histogram.

Normality testing histogram

Figure 1 – Testing for normality using a histogram

The sample contains 20 data elements. To make sure that the intervals in the histogram are equal and consistent, we first standardize the data points (in column C) as in Expectation. E.g. the formula in cell C4 is =STANDARDIZE(B4,B24,B25). Choosing bins from -2 to 2 standard deviations, we create a histogram as described in Histograms.

As you can see from Figure 1, the histogram doesn’t look particularly normal.

QQ Plot

A PP plot (point-point plot) is simply a scatter diagram comparing two samples of the same size. The more similar the underlying distributions, the more closely the scatter points will conform to a line with slope 1. If the data is standardized then the scatter points would be close to the line y = x.

We also use PP plots to compare a data set with a distribution. If the distribution has cdf F(x) and the data set has elements x1, …, xn in ascending order, then the PP plot is the scatter diagram of the set {F(x1), …, F(xn)} versus the set {1/2n, 3/2n, …, 1−1/2n}. Here the second set is an attempt to divide the interval between 0 and n into n evenly spaced intervals (except for the first and last elements which are half the length).

A QQ plot (quantile-quantile plot) is also used to compare a data set with a distribution, and consists of a scatter diagram of the data set {x1, …, xn} in ascending order with the values {F-1(1/2n), F-1(3/2n), …, F-1(1−1/2n)}. Here the ith value F-1(i/n−1/2n) in the second set is the inverse of the cdf at i/n−1/2n (these are the quantiles).

As for the PP plots, if the points on the scatter plot align with the diagonal line y = x then the data set conforms with the distribution.

When using a QQ plot to see whether a data set is normally distributed, you create a scatter diagram between range R1 consisting of the elements x1, …, xn in ascending order and R2 consisting of the values NORM.INV(1/2n, , s), …, NORM.INV(1−1/2n, , s), where = AVERAGE(R1) and s = STDEV(R1).

Alternatively, you can create a scatter diagram between range R1 consisting of the standardized elements z1, …, zn, where each zi = STANDARDIZE(xi, , s), and R2 consisting of the values NORM.S.INV(1/2n), …, NORM.S.INV(1−1/2n).

Commonly, the QQ plot is used much more often than the PP plot. PP plots tend to magnify deviations from the distribution in the center, QQ plots tend to magnify deviation in the tails.

Example 2: Using a QQ plot determine whether the data set with 8 elements {-5.2, -3.9, -2.1, 0.2, 1.1, 2.7, 4.9, 5.3} is normally distributed.

The mean of this data set is .375 and the standard deviation is 3.89. If the data set is normally distributed then for any value x, the cumulative distribution at x would be given by

F(x) = NORMDIST(x, .375, 3.89)

We now split the interval (-∞, ∞) into 8 sub-intervals (-∞, y1 ), (y1, y2), …, (y7, y8), (y8, ∞) such that the area under the standard normal curve for the 2nd through 7th intervals are equal and the area under the curve of the first and last intervals are half the middle intervals. This is equivalent to finding points z1, z3, z5, z7, z9, z11, z13 and z15 such that zi = NORMSINV(i/16). Thus yi = z2i-1. If the original data is normally distributed then

F(xi) = NORMSINV((2i–1)/16).

We summarize this approach in Figure 2, where we have also standardized the original data so that it is easier to compare the standardized data with the standard normal approximation for each data point (under the assumption that the original data was normally distributed). Finally we have included a scatter diagram (the QQ plot) of the data vs. the standardized normal data.

QQ plot testing normality

Figure 2 – Using a QQ plot to test for normality

Cell E5 contains the Excel formula =2*COUNT(A4:A11), while cell E10 contains the formula =NORMSINV(C10/E5) and cell F10 contains =STANDARDIZE(D10,D$6,D7), and similarly for the other cells in columns E and F.

We can see that the data pretty well fits with the trend line, which is a good indicator that the original data is roughly normal. In fact, if the original data is normally distributed, then when the standardized data is plotted against the standard normal values the trend line should be the line .

Real Statistics Data Analysis Tool: The Descriptive Statistics and Normality data analysis tool contained in the Real Statistics Resource Pack allows you to create QQ plots automatically. We illustrate the use of the QQ Plot data analysis tool in the following example.

Example 3: Determine whether the data in Example 1 is normal by using a QQ plot.

The data is repeated in range A3:A23 of Figure 3. To run the analysis press Ctrl-m and select the Descriptive Statistics and Normality option. Fill in the dialog box that appears as shown in Figure 3, choosing the QQ Plot option, and press the OK button.

QQ Plot dialog box

Figure 3 – QQ Plot dialog box

The output is shown in Figure 4

QQ plot Excel normality

Figure 4 – Using Real Statistics QQ plot to test for normality

This time you can see that the data is not quite so normal.

Box Plots

While box plots can’t actually be used to test for normality, they can be useful for testing for symmetry, which often is a sufficient substitute for normality.

Example 4: Use a box plot to gain more evidence as to whether the data in Example 1 is normally distributed.

To produce the Box Plot, press Ctrl-m and select the Descriptive Statistics and Normality option. Fill in the dialog box that appears as shown in Figure 3, choosing the Box Plot option instead of (or in addition to) the QQ Plot option, and press the OK button. The output is shown in Figure 5.

Box plot symmetry

Figure 5 – Using a box plot to test for symmetry

As we can see from Figure 5, the data is relatively symmetric, and so although as we saw in Example 1 and 3, the data is probably not normally distributed, it does appear to be relatively symmetric, which is sufficient for some of the tests that we would like to use.

15 Responses to Graphical Tests for Normality and Symmetry

  1. Pramod Desai says:

    While judging the QQ plot, what is the criteria that it is hugging the diagonal reasonably well. Is it only visual or there is a mathematical treatment? I have a variable with more than 1000 data values. Its QQ plot appears reasonably well (to me) but the Shapiro Will test declares it to be non-normal. Will be grateful for guidance.

    • Charles says:

      Pramod,
      As far as I am aware of, use of a QQ plot is based on visual inspection. I find Shapiro-Wilk to be the most useful test for normality, but its results should agree reasonably well with the QQ plot.
      If you send me an Excel file with your data and test results, I will take a look at it to see why the results are in disagreement. You can find my email address at Contact Us.
      Charles

    • Pramod Desai says:

      I am extremely sorry I did not see this reply from you and hence had raised a fresh query on the same subject. It is very kind of you to have explained the subject and offered to have a look at my data. I will forward the same.

      You have created a great and very useful website on the subject.

      • Charles says:

        Pramod,

        Thank you for sending me your data. Let me make the following observations:

        1. You are testing for the normality of three samples. The Shapiro-Wilk result is for each of the three samples, while the QQ plot that you have created combines all the data into one sample. You need to create three separate QQ plots. The result won’t be that different, although the QQ plot for sample 1 looks a little less normal than for sample 2.

        2. There are a lot of repetitions in the data and so it is a bit difficult for me to determine normality from looking at the QQ plots.

        3. While, in my opinion, in general the Shapiro-Wilk test is the best test for normality, it has a weak spot, namely when there are a lot of repetitions. In this case it is better to use the D’Agostino-Pearson test. The p-values for this test for the three samples are .068, .533 and .541. This would indicate that for each sample you can’t reject the null hypothesis that the sample is normally distributed. Clearly, the result for sample 1 is more marginal than for samples 2 and 3 (i.e. there is a higher probability of a type I error).

        Charles

  2. Sugihartono says:

    I’m very happy to learning from your website, amazing explanation, easy to understand. Great website!

    I have a problem about this statement:
    A QQ plot (quantile-quantile plot) is a PP plot where the samples points are equally spaced. Given a sample of size n which is normally distributed, if we sort the data elements in ascending order then “the area under the normal curve between each point will be equal.”

    the area under the normal curve between each point will be equal, is that means the probability between each point will be the same? I’m try to calculate the area between each point with different between F(x) and can’t get equal values. Am I wrong?

    • Charles says:

      Yes, these paragraphs have not been expressed properly. I have just updated the text to express the concepts more clearly and more accurately
      Thanks for identifying this problem.
      Charles

  3. Jot says:

    I’m trying to plot a QQ plot but I don’t understand how the interval is caluclated in Example 2, Figure 2?

    • Charles says:

      If the sample has n elements then the intervals are based on the endpoints 1, 3, 5, …, 2n-1; i.e. the odd numbers from 1 to 1 less than 2n.
      Charles

  4. MAM says:

    I am not sure what I am mising, but when drawing the standard normal, I do not get a straight line – using excel. I am assuming that the entire point is to standardize the actual data and plot it over what a normal distribution (with same mean and SD) would look like. Am I wrong?

    • Charles says:

      Are you trying to create a QQ Plot as in Figure 4 of the referenced webpage? If so you should get a straight line when using data from a standard normal distribution. If you send me your data I will try to see what went wrong.
      Charles

  5. Afke says:

    Normally I use spss for statistic work, but the company i do my internship don’t have spps. So I must do it with Excel. So I found this helpful website.

  6. Afke says:

    this formula:
    F10 contains the formula =STANDARDIZE(B4,B$15,B$16),

    isn’t right, because your B cells are empty

    • Charles says:

      Thanks Afke for catching this error. I must have changed the worksheet and forgot to change the reference on the webpage. The correct formula is =STANDARDIZE(D10,D$6,D$7). I have now corrected this on the referenced webpage. Charles

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