The normal distribution

Introduction to Statistics in R

Maggie Matsui

Content Developer, DataCamp

What is the normal distribution?

Density function of normal distribution

Symmetrical

Dashed vertical line down the middle of normal distribution

Area = 1

Normal distribution with area underneath curve shaded

Curve never hits 0

Normal distribution with arrows pointing to tails on either side

Described by mean and standard deviation

Mean: 20

Standard deviation: 3

Normal distribution with mean 20 and sd 3

Standard normal distribution

Mean: 0

Standard deviation: 1

Normal distribution with mean 0 and sd 1

Described by mean and standard deviation

Mean: 20

Standard deviation: 3

Normal distribution with mean 20 and sd 3

Standard normal distribution

Mean: 0

Standard deviation: 1

Normal distribution with mean 0 and sd 1

Areas under the normal distribution

68% falls within 1 standard deviation

Normal distribution with area between -1 and 1 highlighted, labeled with 68%

Areas under the normal distribution

95% falls within 2 standard deviations

Normal distribution with area between -2 and 2 highlighted, labeled with 95%

Areas under the normal distribution

99.7% falls within 3 standard deviations

Normal distribution with area between -3 and 3 highlighted, labeled with 99.7%

Lots of histograms look normal

Normal distribution

Standard normal distribution

Women's heights from NHANES

Histogram of women's heights

Mean: 161 cm Standard deviation: 7 cm

Approximating data with the normal distribution

Normal curve drawn over the histogram of women's heights

What percent of women are shorter than 154 cm?

Normal curve drawn over the histogram of women's heights with area less than 154 shaded

16% of women in the survey are shorter than 154 cm

pnorm(154, mean = 161, sd = 7)

0.159

What percent of women are taller than 154 cm?

Normal curve drawn over the histogram of women's heights with area to the right of 154 shaded

pnorm(154, mean = 161, sd = 7, 
      lower.tail = FALSE)

0.8413447

What percent of women are 154-157 cm?

Area less than 157 minus area less than 154

pnorm(157, mean = 161, sd = 7) - pnorm(154, mean = 161, sd = 7)

What percent of women are 154-157 cm?

Area less than 157 minus area less than 154 equals area between 154 and 157

pnorm(157, mean = 161, sd = 7) - pnorm(154, mean = 161, sd = 7)

0.1252

What height are 90% of women shorter than?

Area less than 170 shaded, labeled 90%

qnorm(0.9, mean = 161, sd = 7)

169.9709

What height are 90% of women taller than?

Area greater than about 162 shaded, labeled 90%

qnorm(0.9,
      mean = 161,
      sd = 7,
      lower.tail = FALSE)

152.03

Generating random numbers

# Generate 10 random heights
rnorm(10, mean = 161, sd = 7)

159.35 157.34 149.85 156.75 163.53 156.33 157.22 171.44 158.10 170.12

Let's practice!

Introduction to Statistics in R

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