z-test Normal Distribution

1. “Normal Distribution”
Prabhat Dhakal
M.Sc. III Semester
Food and Industrial Microbiology
National College

2. How data can be distributed ?
1 2 3

3. Data can be distributed around the central
value called a normal distribution

4. Normal distribution
● It is a probability distribution that is symmetric
about the mean
● Also known as Gaussian distribution
● Data near the mean are more frequent than the
data far from the mean
● On plotting histogram, it appears as Bell
shaped curve
E.g.
examining the actual volume of milk in packets
distributed over the market

5. Properties of normal distribution
 Symmetry about the mean
 Unimodal; maximum frequency at
central point
 Asymptotic; approach to axis but do
not touch
 Mean=median=mode

6. Mean Standard
Deviation
Parameters of normal distribution
𝝁 =
𝜮𝒙
𝒏
𝝈 𝒑 =
𝚺 𝒙 − 𝝁 𝟐
𝒏
𝒐𝒓 𝝈(𝒔) =
𝚺(𝒙 − 𝝁)𝟐
𝒏 − 𝟏

7. Standard deviation
● It is a measure of how data are deviated from the mean value
1
2
3

8. Z-score
● It is also called as standard score variant
● measured in terms of standard deviation from
the mean.
● If Z-score is 0, it means the data score (X)is
identical to the mean (𝜇)
● If Z-score is 1 , it indicates data is 1SD from the
mean
● + Z score represents value is above the mean
and
● - Z score represents value is below the mean

9. Conditions :_
1. When data (x)= mean 𝜇
𝑧 =
𝑋−𝜇
𝜎
𝑧 =
𝜇−𝜇
𝜎
𝑧 = 0
2. When data (x)= 𝜇+1𝜎
𝑧 =
𝑋−𝜇
𝜎
𝑧 =
𝜇+1𝜎 −𝜇
𝜎
𝑧 = 1
3. When data (x)= 𝜇+2𝜎
𝑧 =
𝑋−𝜇
𝜎
𝑧 =
𝜇+2𝜎 −𝜇
𝜎
𝑧 = 2
4. When data (x)= 𝜇+3𝜎
𝑧 =
𝑋−𝜇
𝜎
𝑧 =
𝜇+3𝜎 −𝜇
𝜎
𝑧 =3

10. How to calculate Z-score ?
𝑧 =
𝑋 − 𝜇
𝜎
Here,
Z= z-score (standard score)
X= data to be standardized
𝜇=mean
𝜎=standard deviation