Why does data need to be normally distributed

The disadvantage of this approach is that the histogram may change based on the bin widths and there may be bias on how different people may interpret similar graphs especially when there is departure from normality. The statistical way to check if the data is normally distributed is to perform the Anderson-Darling test of normality. In this approach, the data points are used to compute a test statistic A which measures the distance between the expected distribution and the actual distribution.

If this statistic is greater than a certain critical value then the normality of the data is rejected. The test statistic, A, can also be converted into a P value. If the P value is less than alpha default 0. Ideally, we need at least data points before we can check if the data is normally distributed.

The data points show the time to drive to work in minutes for the last month: 30, 42, 28, 32, 25, 29, 27, 31, 38, 36, 31, 29, 27, 26, and We want to check if the data is normally distributed. Do you think the data is normally distributed? Though the histogram follows the blue curve to some extent, it does not closely follow the curve. Any conclusions we draw here are purely qualitative rather than quantitative.

The results are shown below. From this analysis, we can see that the data is close to the normal distribution look at the blue data points that lie close to the red normal distribution curve. If we look at the P value, we can also conclude that since it is less than the default value of alpha 0.

Account Login Login for account management and e-learning. Email Password Check. Remember Me Reset Password. Log me in. Create Account Create a free account to save your e-learning status. The empirical rule in statistics allows researchers to determine the proportion of values that fall within certain distances from the mean.

The empirical rule is often referred to as the three-sigma rule or the The empirical rule allows researchers to calculate the probability of randomly obtaining a score from a normal distribution. This means there is a Statistical software such as SPSS can be used to check if your dataset is normally distributed by calculating the three measures of central tendency.

If the mean, median and mode are very similar values there is a good chance that the data follows a bell-shaped distribution SPSS command here. Normal distributions become more apparent i. You can also calculate coefficients which tell us about the size of the distribution tails in relation to the bump in the middle of the bell curve.

McLeod, S. Introduction to the normal distribution bell curve. Toggle navigation. Saul McLeod , published What are the properties of the normal distribution? What is the difference between a normal distribution and a standard normal distribution?

The bell-shaped curve is a common feature of nature and psychology. Edit: To clarify, I am not making any claims about how much real world data is normally distributed. I am just asking about theorems that can give insight into what sort of processes might lead to normally distributed data.

Many limiting distributions of discrete RVs poisson, binomial, etc are approximately normal. Think of plinko. In almost all instances when approximate normality holds, normality kicks in only for large samples.

Most real-world data are NOT normally distributed. A paper by Micceri called " The unicorn, the normal curve, and other improbable creatures " examined large-scale achievement and psychometric measures. He found a lot of variability in distributions w. In a paper by Steven Stigler called " Do Robust Estimators Work with Real Data " he used 24 data sets collected from famous 18th century attempts to measure the distance from the earth to the sun and 19th century attempts to measure the speed of light.

He reported sample skewness and kurtosis in Table 3. The data are heavy-tailed. In statistics, we assume normality oftentimes because it makes maximum likelihood or some other method convenient. What the two papers cited above show, however, is that the assumption is often tenuous. This is why robustness studies are useful. There is also an information theoretic justification for use of the normal distribution.

Given mean and variance, the normal distribution has maximum entropy among all real-valued probability distributions. There are plenty of sources discussing this property. A brief one can be found here. A more general discussion of the motivation for using Gaussian distribution involving most of the arguments mentioned so far can be found in this article from Signal Processing magazine.

In physics it is CLT which is usually cited as a reason for having normally distributed errors in many measurements. The two most common errors distributions in experimental physics are normal and Poisson. The latter is usually encountered in count measurements, such as radioactive decay. Another interesting feature of these two distributions is that a sum of random variables from Gaussian and Poisson belongs to Gaussian and Poisson.

The CLT is extremely useful when making inferences about things like the population mean because we get there by computing some sort of linear combination of a bunch of individual measurements.

However, when we try to make inferences about individual observations, especially future ones eg , prediction intervals , deviations from normality are much more important if we are interested in the tails of the distribution.

For example, if we have 50 observations, we're making a very big extrapolation and leap of faith when we say something about the probability of a future observation being at least 3 standard deviations from the mean. Sign up to join this community.