Download The Normal Distribution (1)

Subject: MDM4UI
Topic: The Normal Distribution (1)
Unit: 6.5
Homework: Page 176
1, 3b, 6, 9a
Objective:
To look at the most famous distribution curve.
Notes:
Normal distributions are symmetrical and approach zero at the extremes
Of the data, 68% is within one standard deviation of the mean, 95% is within two standard
deviations of the mean, and 99.7% is within three standard deviations of the mean.
The mean, median, and mode are all equal and fall on the line of symmetry of the distribution
The area under any normal curve is 1. The percent of the data that lies between two values in a
normal distribution is equivalent to the area under the normal curve between these values.
The notation used to describe a normal distribution, of the variable X, is X ∼ N ( x , σ 2 ) .
Examples:
Giselle is 168cm tall. In her high school, boys’ heights are normally distributed with a mean of
174cm and a standard deviation of 6cm. What is the probability that the first boy Giselle meets at
school tomorrow will be taller than she is?
Solution:
We need to find the probability that the height of the first boy falls into a certain range. The
normal distribution is a continuous curve, so the probability is the area under the appropriate part
of the probability-distribution curve.
Giselle’s Height
6
168
174
The shaded red colour represents the P(168<Boys height)
We need to find P(168< boy’s height). For this distribution, 168 = µ − σ . So, you can use the
fact that, for any normally distributed random variable X, P ( µ − σ < X < µ + σ ) 68%
Since 168 is exactly one standard deviation (6) form 174, the are under curve between 168 and
174 is 34%. The are under curve from 174 to infinity is 50&.
Therefore the red area is 84%.
The probability is 84%.