Maths for Engineers and Scientists 1

Maths for Engineers and Scientists 1

The Normal Distribution

• The normal distribution is the most important continuous distribution used in statistics. It is represented by the classic bell shape shown in figure below:

The normal distribution has the following properties:

• The normal distribution has the following properties:
• 1) Its graph is a bell-shaped curve
• 2) The curve is symmetric about the vertical centerline. This centerline passes through the value that is the mean, the median and the mode of the distribution.

3) The normal distribution is completely determined when its mean (or ) and its standard deviation are known.

• 3) The normal distribution is completely determined when its mean (or ) and its standard deviation are known.
• 4) The total area under the normal curve is 1. The area under the curve from value to value represents the percentage of the scores that lie between and . Thus the area under the curve from to represents the probability that a score chosen at random will lie between and .

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Normal distribution is also known as Gaussian distribution. The Normal distribution is given by

• Normal distribution is also known as Gaussian distribution. The Normal distribution is given by

The constant parameters and are the mean and standard deviation, respectively, for the Normal distribution, which is denoted N( ; ).

If the mean =0 and standard deviation is =1, then the normal distribution is called standard normal distribution.

• If the mean =0 and standard deviation is =1, then the normal distribution is called standard normal distribution.
• Example 1. Given a standard normal distribution N( ; 1). What is the probability that
• a) is less than 1.38
• b) is greater than 1.82
• c) is between 1.23 and 1.87
• d) is less than 1.27 or greater than 1.94?
• E) What is the value of if only 2.5% of all possible values are larger?
• F) Between what two values of (symmetrically distributed around the mean) will 68.26 % of all possible values be contained?

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It is possible to transform any normal distribution ; ) to the standard normal distribution. To transform any given value of on ; ) to its equivalent -value on use the formula

• It is possible to transform any normal distribution ; ) to the standard normal distribution. To transform any given value of on ; ) to its equivalent -value on use the formula

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Example 2. Given a normal distribution with =100 and =10, what is the probability that

• Example 2. Given a normal distribution with =100 and =10, what is the probability that
• a)
• b)
• c) or
• D) Between what two values of (symmetrically distributed around the mean) are 80 % of all possible values contained?
• Example 3. A set of final examination grades in an introductory statistics course is normally distributed with a mean of 75 and a standard deviation of 7.
• What is the probability of getting a grade of 90 or less on this exam?
• What is the probability that a student scored between 60 and 87?
• The probability is 6% that a student taking the test scores higher than what grade?

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Example 4. A company determined that on an annual basis the distance traveled per truck is normally distributed with a mean of 55 thousand miles and a standard deviation of 14 thousand miles.

• Example 4. A company determined that on an annual basis the distance traveled per truck is normally distributed with a mean of 55 thousand miles and a standard deviation of 14 thousand miles.
• What proportion of trucks can be expected to travel between 38 and 62 thousand miles in the year?
• What percentage of trucks can be expected to travel either below 46 or above 71 thousand miles in the year?