The Normal Distribution

Section 8.2

The Galton Board

• Developed in the late 19th century by Sir Francis Galton, a cousin of Charles Darwin

• Theorized that with enough pegs in the board and with a large enough number of marbles, this discrete binomial distribution would come closer and closer to a continuous curve he referred to as the Bell Curve, or Normal Distribution

The Normal Distribution

• Galton theorized that this continuous distribution could describe many measurable statistics

• Eg) Human height, human weight, human incomes, the number of hairs on your head, the mean cost of bread over time, etc.

Turns out… it does• Galton also determined that according to this model, the

following was always true:

Suppose we know the following…

• 1. What percentage of men are less than 170 cm?

• 2. What percentage of men are between 160 and 180 cm?

• 3. What percentage of men are between 150 and 190 cm?

Human male heights are normally distributed with a mean of 170 cm and a standard deviation of 10 cm.

… and for women

• 1. What percentage of women are between 138 and 182 cm?

• 2. What percentage of women are taller than 149 cm?

• 3. If you are a female who is 193 cm tall, what percentile are you? (ie what percentage of the population is shorter than you)

Human female heights are normally distributed with a mean of 160 cm and a standard deviation of 11 cm.

University Marks

• 1. What percentage of students are scoring less than 55%?

• 2. You are scoring 70%. What percentile are you in?

• 3. Approximately what percentage of the class is scoring more than 60%?

In a first year University Chemistry class, the marks are normally distributed with a mean of 40% and a standard deviation of 15%

“Belling” or “Curving” dataA class average is at 35% with a standard deviation of 17%.The professor wants to ‘curve’ these marks, and have a class mean of

65% with a standard deviation of 10%.1. If you were scoring 35% in the original class, what would you be

getting in the new class?

1. If you were scoring 52% in the original class, what is your new mark?

1. You scored 18% in the original class, what is your new mark?

4. Approximately, what is your new mark if your old mark was 46%?

Summarize:• Where did the Normal Distribution originally

come from?

• Why is the normal distribution useful?

• Explain the process behind ‘curving’ data

Z-Scores• A z-score is used to measure how many

standard deviations you are above or below the mean.

• Example) The mean is 50% and the standard deviation is 10%. What is the z-score of 60%?

• What is the z-score of 40%?

• How about the z-score of 65%?