12-Normal Distribution

Data can be “distributed” (spread out) in different ways.

But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a “Normal Distribution” like this:

The Normal Distribution has:

• mean = median = mode
• symmetry about the center
• 50% of values less than the mean and 50% greater than the mean

Standardization and z-score

Standardization

It is good to know the standard deviation, because we can say that any value is:

• likely to be within 1 standard deviation (68 out of 100 should be)
• very likely to be within 2 standard deviations (95 out of 100 should be)
• almost certainly within 3 standard deviations (997 out of 1000 should be)

So to convert a value to a Standard Score (“z-score”):

• first subtract the mean,
• then divide by the Standard Deviation

z-score formula

Example :- A survey of daily travel time had these results (in minutes):

26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34

The Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes

Convert the values to z-scores (“standard scores”).

To convert 26:

first subtract the mean: 26–38.8 = -12.8,

then divide by the Standard Deviation: -12.8/11.4 = -1.12

So 26 is -1.12 Standard Deviations from the Mean

And here they are graphically:

Credits:- Mathsisfun.com