# Normal Distribution

** Published:**

This post covers Normal Distribution.

Ztable: https://www.mathsisfun.com/data/standard-normal-distribution-table.html

Ztable: Z

Different ways that data can be distributed

Examples of Normal Distribution

- Heights of people
- size of things produced by machines
- errors in measurements
- blood pressure
- marks on a test

## Normal Distribution

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

## Standard Deviation

- measure of how spread out numbers are
- square root of the Variance
Variance is average of the

**squared**differences from the Mean- Example
- Heights: 600mm, 470mm, 170mm, 430mm and 300mm
- Compute Mean, the Variance, and the Standard Deviation
- Mean
- 394

- Variance
- Each Dogâ€™s Difference from the mean
- 21704

- Standard Deviation
- 147.32
- SD is useful since we can show which heights are within one Standard Deviation (147) of the mean (394 mm)
- Using Standard Deviation, we have a standard way of knowing what is normal and what is extra large, or extra small

- Correction for Sample Data
- If the data is population, then variance is average of squared differences
- If the data is sample from a bigger population, we divide by N-1 for calculating variance
- Sample Variance: 27130
- Sample Standard Deviation: 165

- Example
- 95% of students are between 1.1m and 1.7m tall. Assume data is normally distributed, compute mean and standard deviation
- Mean is halfway between 1.1m and 1.7m
- Mean = (1.1m + 1.7m) / 2 = 1.4m

- 95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:
- $1~SD = \frac{1.7m-1.1m}{4}=0.15$

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)

## Standard Scores

- Example
- One student is 1.85m tall
- 1.85m is
**3 standard deviations**from the mean of 1.4- $ \frac{1.85 - 1.4}{.15} = \frac{.45}{.15}=3 $

- Thus, z-score is 3.0

- Example
- 26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34
- Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes

- Why Standardize?
- Marks out of 60
- 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17

- Mean = 23 and Standard Deviation = 6.6
- -0.45,
**-1.21**, 0.45, 1.36, -0.76, 0.76, 1.82,**-1.36**, 0.45, -0.15, -0.91- Only two lower than one SD

- Marks out of 60

- Your score in a recent test was 0.5 standard deviations above the average, how many people scored lower than you did?
- Between 0 and 0.5 is
**19.1%** - Less than 0 is
**50%**(left half of the curve) - So the total less than you is:
- 50% + 19.1% = 69.1%

- Between 0 and 0.5 is

### References

- https://www.mathsisfun.com/data/standard-normal-distribution.html
- https://www.statisticshowto.com/probability-and-statistics/normal-distributions/