Lesson 3: Normal Distribution

Question 1

Distribution

Answer 1

• histogram of sample space (all possible sample values)
• X-axis: all possible values in sample space
• Y axis: Frequency of the sample value

Question 2

Standardized values

z value

Answer 2

• Compute Probabilities
• Compare two different distributions
• We compute standardized values
• z = 2 : Data value is 2 standard deviations above the mean
• z = -1.6 : Data value is 1.6 standard deviations below the mean

z = (Data value (y) - mean(μ)) / Standard deviation (σ)

Question 3

p-value

Answer 3

• Represents the area under the normal distribution curve towards left side
• 1.0 z-value = 0.15 + 2.35 + 13.50 + 34.00 + 34.00 = 84% p-value (0.8413)
• area under normal distribution curve = 1

Question 4

p-value calculation for specific range

example

Answer 4

probability of student scoring b/n 450 and 600 on SAT
mean = 500, sd = 100
z = (600-500)/100 = 1.0 = p-value 0.8413
z = (450-500)/100 = -0.50 = p-value 0.3085
0.8413 - 0.3085 = 0.5328 or 53.28%

Question 5

Standard normal curve
(𝜇=0, 𝜎=1)

Answer 5

• Symmetric about its mean 𝜇=0,𝜎=1
• Mean = Median = Mode
• Single peak at z=0
• Inflection point at −1𝑎𝑛𝑑+1
• Area under the curve = 1
• Area of left ( 𝑚𝑒𝑎𝑛 𝜇=0) = Area of right = ½
• Follows the Empirical Rule

Question 6

Normalizing Data

Computing z-values

Excel and R

Answer 6

• Excel: =STANDARDIZE(data value, mean, sd)
• R: scale(data, mean, sd)

Question 7

Computing p-values
𝑃(𝑧)𝑥 < 𝑧

Excel and R

Answer 7

‘Left’ Area (Probability) under Standard Normal Curve
- Excel: =NORMSDIST(z-value) Normal standard dist
- R: pnorm(z-value) Cumulative Density Function (cdf)

‘Right’ Area (Probability) under Standard Normal Curve
- Excel: =1-NORMSDIST(z-value)
- R: 1 - pnorm(z-value)

‘In Between’ Area (Probability) under Standard Normal
Curve
- Excel: =NORMSDIST(high z) - NORMSDIST(low z)
- R: pnorm(high z) - pnorm(low z)

Question 8

Converting p-value (‘left’ area) to z-value

Excel and R

Answer 8

• Excel: =NORMSINV(p-value)
• R: qnorm(p-value) Quantiles

Question 9

Relative Frequency
(Histogram)

Answer 9

• Individual frequency / total frequencies
• Histogram y-axis = Density
• frequency total area = 1.0 (100%)
• aka Probability Distribution Function (PDF)

Question 10

Uniform Distribution Function

Answer 10

• When the probability of all possible events in the sample space is same
• eg. 6 dice sides = equal probability
• R: runif(n=?) = generate Random Uniform Distribution

Question 11

Continuous Distributions

types (12)

Answer 11

1. Uniform
2. Normal
3. Chi Square
4. Fisher’s F
5. Student’s t
6. Gamma
7. Exponential
8. Beta
9. Cauchy
10. Lognormal
11. Logistics
12. Weibull

Question 12

Discrete Distributions

types (5)

Answer 12

1. Binomial
2. Poisson
3. Hypergeometric
4. Negative Binomial
5. Wilcox

Question 13

Normal Distribution

properties

Answer 13

• Symmetric about its mean 𝜇
• Mean = Median = Mode
• Single peak at 𝑥=𝜇
• Inflection point at 𝜇−𝜎𝑎𝑛𝑑𝜇+𝜎
• Area under the curve = 1
• Area of left ( 𝑚𝑒𝑎𝑛 𝜇) = Area of right = ½
• Follows the Empirical Rule

Question 14

Density Distribution

R function

Answer 14

dnorm()

Question 15

Random Numbers Distribution

Answer 15

• R: rnorm(n= ?, mean= ?, sd= ?)
• generates n numbers normally distributed with specified mean and sd
• clean histogram and QQ plot if n is big enough

Question 16

Testing Normality

Histogram

Answer 16

• should see bell-shaped curve if normal distribution
• changes shape with different bin sizes = not as accurate
• never sure if you have right bin size

Question 17

Testing Normality

Quantile-Quantile (QQ) Plot

Answer 17

• Data is plotted against a theoretical normal distribution
• If you see a straight line, data is normally distributed
• X-axis: quantile data (theoretical normal dist)
• Y-axis: sorted data

Procedure:
1. Sort data
2. Divide standard normal curve into n+1 portions
3. Take z-value for each portion
4. Plot z-values on x-axis and sorted data on y-axis
2. Plot against appropriate quantiles from standard normal distribution

Question 18

QQ Plot

Calculating z-values

Answer 18

1. assign i values of each sorted point (eg. i = 1,2,3,etc)
2. p-value for each = i / (n+1)
3. calculate z-values from p-values (Excel or R)

Question 19

QQ Plot generation

R

Answer 19

qqnorm(vector)
qqline(vector)

Question 20

R commands

sort column values

ascending

Answer 20

table[order(table$column),]

Question 21

R commands

histogram frequency counts

Answer 21

• variable = hist(table$column,seq(start, end, interval))
• variable$count

Question 22

R commands

cumulative sum linear plot

Answer 22

-adds up all vector values
- plot(cumsum(variable$counts),type=’l’)

Question 23

R commands

calculate boxplot stats

Answer 23

• variable = boxplot(table$column)
• outlier = variable$out
• Min = variable$stats[1]
• 1st Quartile = variable$stats[2]
• Median = variable$stats[3]
• 3rd Quartile = variable$stats[4]
• Max = variable$stats[5]

Question 24

R command

filtering results

Answer 24

new.variable = (Table$Column > value) & (Table$Column < value)