- 1 What is a Normal Distribution?
- 2 Why Normal Distribution is Important?
- 3 Properties of a Normal Distribution
- 4 Standard Normal Distribution
- 5 Standardization of a Normal Distribution
- 6 What is a Z score?
- 7 Inference of a Z Score
- 8 Percentile/Area under the Z distribution/CDF Calculation
What is a Normal Distribution?
A Normal Distribution is a kind of continuous probability distribution wherein most of the values cluster in the middle of the range and there are lesser values towards the extreme ends of the range. You can intuitively appreciate this fact by looking at the Normal distribution graph.
Mathematically a Normal distribution is given as below.
Why Normal Distribution is Important?
Many of the real-life events approximately follow Normal distribution. A few examples are –
- People’s heights and weights
- Population’s blood pressure
- Test Scores
- Measurement errors
- Noise in a signal
Properties of a Normal Distribution
- All Normal distributions are symmetrical around mean.
- The mean, the median, and the mode of the normal distribution are the same.
- 68.27% of the values in a normal distribution lie within one standard deviation.
- 95.45% of the values in a normal distribution lie within 2 standard deviations.
- 99.73% of the values lie within 3 standard deviations from the mean.
- In continuous distribution, we always talk about the probability of a range of values. The probability of a specific outcome is always zero.
Standard Normal Distribution
- mean = median= mode =0
- Standard deviation = 1
- Area under the Standard Normal Distribution curve is equal to 1.
Standardization of a Normal Distribution
What is a Z score?
where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution, and σ is the standard deviation of the original distribution.
If all the values in a distribution are transformed to Z scores, then the distribution will have a mean of 0 and a standard deviation of 1. This process of transforming a distribution to one with a mean of 0 and a standard deviation of 1 is called standardizing the distribution.
An example of standardization of a Normal Distribution is given in the following picture. In this example, it is given that the mean = 8, standard deviation = 3 and a particular value in normal distribution is x=6. We want to know what percentage of data lie at x<6 (that is percentile).
Inference of a Z Score
- Within one standard deviation, 68.27% values lie. The probability that value lies within Z=-1 to +1 is 68.27%
- Within two standard deviations, 95.45% values lie. The probability that value lies within Z=-2 to +2 is 95.45%
- Within three standard deviations, 99.73% values lie. The probability that value lies within Z=-3 to +3 is 99.73%
Percentile/Area under the Z distribution/CDF Calculation
- Calculate Z score using the formula Z=(x-μ)/σ ( in last example Z=-0.67)
- Once the Z score is determined, percentile or area under standard normal distribution can be calculated as below.
This video explains Normal Distribution concepts briefly.