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I'm a newbie here. My question is the following.

Are the following set of values normally distributed? 26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34

The above values are from the below link https://www.mathsisfun.com/data/standard-normal-distribution.html

They go on to compute the mean and standard deviation and the corresponding z scores assuming they are normally distributed.

However when i plotted the values on a histogram using excel, i get the following chart(Attached image) which shows a positive skewness and we know that a normally distributed set of observations has no skewness at all i.e its perfectly symmetrical.

Do we need to transform the data-set into normally distributed values before calculating the mean , standard deviation and the z scores ? ...since in real world situations , data-sets may not be normally distributed , then how do we go ahead to perform statistical tests on them.enter image description here

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2 Answers 2

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For two reasons you picked the wrong kind of plot for visualizing your sample. First, you assume that your data is continuous, so there is no point in counting distinct values. Second, your sample is very small, so even with discrete numbers, in most cases you can expect small counts per value that result with a flat barplot.

Recall that for continuous random variable $\Pr(X=x)=0$, so assuming that we are talking about continuous random variable we would rather not expect different values to appear in your sample multiple times -- so counting their occurrences is misleading. That is why, for continuous random variables we use probability densities, i.e. probabilities "per foot". Instead of counting how many number each of the numbers appeared, you should count their counts in intervals. That is why for visualizing your data rather than using bar plot, you should use histogram, or density plot.

Since your sample is very small, histogram could be misleading because there is limited number of bars that can be used and small number of cases that will fall into each of the bars (no matter if your variable is discrete or continuous). In this case, density plot (see below) could be more informative.

enter image description here

As a counter-example, below you can see barplot of values generated from normal distribution using pseudo-random numbers generator (black bars) and density plot (red line).

enter image description here

As you can see, barplot would "suggest" that this perfectly normal data is almost uniformly distributed...

As about if your sample is normally distributed -- it seems that the data contains of integers rather than real numbers, so obviously it is not perfectly normal. Moreover, the distribution is skewed rather than symmetric. However in most cases this is not a problem because we are interested in approximate normality. See: Is normality testing 'essentially useless'?

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  • $\begingroup$ In the first paragraph you imply that these data are continuous while in the last paragraph you say they are discrete... $\endgroup$
    – amoeba
    Mar 29, 2016 at 9:33
  • $\begingroup$ @amoeba good point, hope that adding the word "assuming" makes it more clear ;) $\endgroup$
    – Tim
    Mar 29, 2016 at 9:39
  • $\begingroup$ I still don't understand why you start with saying that "You picked the wrong kind of plot for visualizing your sample"; if the data consist of integers, it's a perfectly fine plot. $\endgroup$
    – amoeba
    Mar 29, 2016 at 10:45
  • $\begingroup$ @amoeba I added a leading paragraph, hope it clarifies things. $\endgroup$
    – Tim
    Mar 29, 2016 at 10:50
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Are the following set of values normally distributed? 26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34

Clearly not; they're integers.

[More properly, it's not a set of observed values that's normally distributed (the ECDF of a set of $n$ known values is discrete, the values themselves are bounded and so on); normality is an attribute of a population distribution from which an observed sample might have been drawn. But not this one.]

However, while it's often clear we cannot have a sample from a normal distribution for one reason or another, rarely is it interesting to ask whether the sample came from a normal distribution. A more relevant question is whether it might be a suitable approximation -- but to answer that question you need to know more about what you're doing, what impact the non-normality you have might have on it, and what your tolerance for that impact might be (or your audience's tolerance, perhaps).

(One thing worth noting about the shape can be seen from a QQ-plot -- or any number of other displays, depending on what you're used to using to investigate distributional shape. You should show a suitable display and interpret it. The display you show -- which is not a histogram in spite of being labelled as one -- is not really suitable, since it disguises the relative gaps in the data. It appears to be treating the x-axis values as a set of ordered category labels rather than something where the number indicates position.)

qqplot of data
Q-Q plot of the data indicates skewness

We know that a normally distributed set of observations has no skewness at all

I sure don't know that; in fact I know it's untrue -- a sample from a normal distribution can certainly be somewhat skewed, just by random variation. It's the population that has no skewness at all.

But your conclusion -- that the data indicate skewness -- is correct, it's just much harder to see in in that chart in your question.

enter image description here

Here's a dotplot, which does a better job than the bar chart. An actual histogram should be adequate. (If there was more data, I'd look at something else -- with separate thin bars representing relative frequency, as your display has, but with the x-position representing the values, akin to a histogram. In R you get this with plot(table(x)), but for very small samples like this with few repeated values I prefer the dotplot.)

Do we need to transform the data-set into normally distributed values before calculating the mean, standard deviation and the z scores?

What could you conclude from the mean (etc) of transformed data?

...since in real world situations, data-sets may not be normally distributed

In real world situations, you don't really have normal distributions, except in a few special situations.

then how do we go ahead to perform statistical tests on them.

  1. Not all tests assume normality

  2. Even for those that do, the assumption of normality is not always very important (sometimes it may matter only a little, sometimes it might matter a lot -- it can depend on the test and on the sample size).

  3. Transformation is frequently not the first thing you should think about doing. You should first really pay attention to what questions you need to ask of the data (what do you need to find out?). Then you can worry about what might be suitable ways to do that. It might involve transformation, but it might much better involve something else.

What are you interested in finding out from these data? If you don't know, why would you transform first? It might have no value in answering the questions of interest.

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