Characterstics of Normal Distribution

 

There are several characteristics or properties of a dataset that indicate that it is normally distributed. Here are some of the key things to look for:

1. Symmetry: A normal distribution is symmetric, which means that the left half of the distribution is a mirror image of the right half. In other words, the mean, median, and mode are all equal and located at the center of the distribution.
2. Bell-shaped curve: A normal distribution has a bell-shaped curve that is relatively smooth and continuous. The curve is highest at the mean and tapers off gradually in both directions.
3. Empirical rule: The empirical rule, also known as the 68-95-99.7 rule, states that approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.
4. Skewness and kurtosis: A normal distribution has zero skewness and zero excess kurtosis, which means that the tails of the distribution are not too heavy or too light compared to the normal distribution.
5. QQ plot: A QQ plot, or quantile-quantile plot, is a graphical method for comparing the distribution of the data to the normal distribution. If the data is normally distributed, the points on the QQ plot will fall along a straight line.
6. Mean, median, and mode are equal: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution. This is often referred to as the central tendency of the data.
7. Probability density function: A normal distribution can be fully described by its probability density function, which is a mathematical function that describes the probability of observing a particular value or range of values in the distribution.
8. Standard deviation: The standard deviation of a normally distributed dataset provides a measure of the spread or variability of the data. About 68% of the data falls within one standard deviation of the mean, and about 95% falls within two standard deviations.
9. Independent, identically distributed (iid) samples: If a sample of data is drawn from a normally distributed population, and the samples are independent and identically distributed, then the sample mean will also be normally distributed.
10. Z-scores: Z-scores, which measure the number of standard deviations a value is from the mean, are commonly used in normal distribution calculations and statistical tests.

It's important to note that not all datasets that exhibit these characteristics are necessarily normally distributed, and there are statistical tests that can be used to confirm normality. However, if a dataset displays all or most of these characteristics, it is a good indication that it is normally distributed.