17 Normal Distribution Concept Overview

NORMAL DISTRIBUTION | CONCEPT OVERVIEW The TOPIC of NORMAL DISTRIBUTION can be referenced on page 39 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

CONCEPT INTRO: The NORMAL DISTRIBUTION refers to a family of continuous probability distributions defined by the normal equation. The parameters of the normal distribution are 𝜇𝜇 𝑎𝑎𝑎𝑎𝑎𝑎 𝜎𝜎.

The FORMULA for NORMAL PROBABILITY DENSITY FUNCTION can be referenced under the SUBJECT of ENGINEERING PROBABILITY AND STATISTICS on page 39 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The normal distribution is a unimodel distribution, such that 𝑥𝑥 = 𝜇𝜇 with two points of inflection (each located at a distance of 𝜎𝜎 to ether side of the mode). The average of 𝑛𝑛 observations tends to become normally distributed as 𝑛𝑛 increases. 𝑓𝑓 𝑥𝑥 =



1

𝜎𝜎 2𝜋𝜋

𝑒𝑒

/

0 2/3 5 1 4 −

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Where: • 𝜇𝜇 is the population mean

• 𝜎𝜎 is the standard deviation of the population • 𝑥𝑥 is a normal random variable • 𝜎𝜎 1 is the standard deviation

The normal equation is the PROBABILITY DENSITY FUNCTION for the normal distribution. The graph of the normal distribution depends on two factors – the mean and the standard deviation. The MEAN OF THE DISTRIBUTION determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distribution looks like a symmetric, bell-shaped curve. There are a few key points to remember when dealing with continuous probability distributions: • The total area under the normal curve is equal to 1.

• The probability that a normal random variable 𝑥𝑥 equals any particular value is 0.

• The probability that 𝑥𝑥 is greater than a certain value 𝑎𝑎 equals the area under the normal curve bounded by 𝑎𝑎 and plus infinity.

• The probability that 𝑥𝑥 is less than a certain value 𝑎𝑎 equals the area under the normal curve bounded by 𝑎𝑎 and minus infinity.



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Every normal curve (regardless of its mean or standard deviation) conforms to the 6895-99.7 rule, that is: • 68% of the area under the curve falls within 1 standard deviation of the mean • 95% of the area under the curve falls within 2 standard deviations of the mean • 99.7% of the area under the curve falls within 3 standard deviations of the mean. To find the probability associated with a normal random variable of a standard normal distribution, we can use the Z-SCORE and the normal distribution table. A z-score follows a standardized normal distribution function. The FORMULA FOR CALCULATING A Z-SCORE can be referenced under the topic of NORMAL DISTRIBUTION (GAUSSIAN DISTRIBUTION) on page 39 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Every normal random variable 𝑥𝑥 can be transformed into a z-score using the equation below:

𝑍𝑍 = Where:

𝑥𝑥 − 𝜇𝜇 𝜎𝜎

• 𝑥𝑥 is the normal random variable • 𝜇𝜇 is the mean

• 𝜎𝜎 is the standard deviation of 𝑥𝑥



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The NORMAL DISTRIBUTION TABLE can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 9.4 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The normal distribution function can not be integrated so we use Z-TABLES or the NORMAL DISTRIBUTION TABLE, which are for the standard normal distribution with 𝜇𝜇 = 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝜎𝜎 = 1.

The normal distribution shows a cumulative probability associated with a particular zscore. Table rows show the whole number and tenths place of the z-score. Table columns show the hundredths place. The cumulative probability appears in the cell of the table. The NOTATIONS ON THE UNIT NORMAL DISTRIBUTION TABLE can be referenced under the topic of NORMAL DISTRIBUTION (GAUSSIAN DISTRIBUTION) on page 39 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A unit normal distribution table is included at the end of the reference handbook section on probability and statistics. In the table, the following notations are utilized: • 𝐹𝐹 𝑥𝑥 = 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − ∞ 𝑡𝑡𝑡𝑡 𝑥𝑥 • 𝑅𝑅 𝑥𝑥 = 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑚𝑚 𝑥𝑥 𝑡𝑡𝑡𝑡 ∞

• 𝑊𝑊 𝑥𝑥 = 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − 𝑥𝑥 𝑡𝑡𝑡𝑡 𝑥𝑥

• 𝐹𝐹 −𝑥𝑥 = 1 − 𝐹𝐹(𝑥𝑥)



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Often times we may not be interested in the probability that a normal random variable falls between minus infinity and a given value. We may want to know the probability that it lies between a given value and plus infinity, or lies between two given values. These probabilities are easy to computer from a normal distribution table. Consider the following: • The probability that a standard normal random variable (𝑧𝑧) is greater than a given value (𝑎𝑎), 𝑃𝑃(𝑧𝑧 < 𝑎𝑎), is determined by 1 − 𝑃𝑃(𝑧𝑧 < 𝑎𝑎).

• The probability that a standard normal random variable lies between two values, 𝑃𝑃(𝑎𝑎 < 𝑧𝑧 < 𝑏𝑏) is 𝑃𝑃(𝑧𝑧 < 𝑏𝑏) − 𝑃𝑃(𝑧𝑧 < 𝑎𝑎).

Often, events in the real world follow a normal distribution. This allows us to use the normal distribution as a model for assessing probabilities associated with these events.



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CONCEPT EXAMPLE: A company pays its employees an average wage of $3.25/hour with a standard deviation of 60 cents. If the wages are approximately normally distributed, the proportion of the workers getting wages between $2.75 and $3.69 an hour is most close to: A. 50%

B. 55% C. 60%

D. 65%



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SOLUTION: The FORMULA FOR CALCULATING A Z-SCORE can be referenced under the topic of NORMAL DISTRIBUTION (GAUSSIAN DISTRIBUTION) on page 39 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Every normal random variable 𝑥𝑥 can be transformed into a z-score using the equation below:

𝑍𝑍 = Where:

𝑥𝑥 − 𝜇𝜇 𝜎𝜎

• 𝑥𝑥 is the normal random variable • 𝜇𝜇 is the mean

• 𝜎𝜎 is the standard deviation of 𝑥𝑥

To analyze this problem, we will calculate the z-score and then use the normal distributions tables, to find probabilities associated with the z-scores. We are given the following values to solve for z-scores of each wage range: • x0 = 2.75

• 𝑥𝑥1 = 3.69 • 𝜇𝜇 = 3.25

• 𝜎𝜎 = 0.60



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For the first group of with wages between $2.75-$3.25 per hour, we will calculate the zscore, 𝑍𝑍0 , of the probability a wage falls in that range as: 𝑧𝑧0 =

(2.75 − 3.25) = −0.833 0.60

For the second group of with wages between $3.25-$3.69 per hour, we will calculate the z-score, 𝑍𝑍1 , of the probability a wage falls in that range as: 𝑧𝑧1 =

(3.69 − 3.25) = 0.733 0.60

Which means that:

𝑝𝑝 2.75 < 𝑥𝑥 < 3.69 = 𝑃𝑃(−0.833 < 𝑧𝑧 < 0.733)

We then look at the normal distribution tables to find the corresponding probability values for each z-score. A unit normal distribution table is included at the end of the reference handbook section on probability and statistics. In the table, the following notations are utilized: • 𝐹𝐹 𝑥𝑥 = 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − ∞ 𝑡𝑡𝑡𝑡 𝑥𝑥

• 𝑅𝑅 𝑥𝑥 = 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 𝑡𝑡𝑡𝑡 ∞

• 𝑊𝑊 𝑥𝑥 = 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 − 𝑥𝑥 𝑡𝑡𝑡𝑡 𝑥𝑥

• 𝐹𝐹 −𝑥𝑥 = 1 − 𝐹𝐹(𝑥𝑥)



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For the first group of with wages between $2.75-$3.25 per hour, we use the z-score, 𝑍𝑍0 = 0.833, to find the probability the wages are LESS THAN $3.25 an hour. We use the

𝐹𝐹(𝑥𝑥) column to indicate 𝑃𝑃 𝑥𝑥 < 3.25 and 𝑥𝑥 = 0.8 row, as this is the closet value we can

use without interpolating. We use the F(x) column as we are looking to calculate the area under the curve from −∞ 𝑡𝑡𝑡𝑡 𝑥𝑥, 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑥𝑥 = 0.8.

The NORMAL DISTRIBUTION TABLE can be referenced under the topic of ENGINEERING PROBABILITY AND STATISTICS on page 9.4 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

𝑃𝑃 𝑥𝑥 < 3.25 = −0.833 = 0.7881



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For the second group of with wages between $3.25-$3.69 per hour, we will calculate the z-score, 𝑍𝑍1 = 0.733, to find the probability the wages are GREATER THAN $3.25 an

hour. We use the 𝐹𝐹(𝑥𝑥) column to indicate 𝑃𝑃 𝑥𝑥 > 3.25 =and 𝑥𝑥 = 0.7 row, as this is the

closet value we can use without interpolating. We use the 𝑅𝑅(𝑥𝑥) column as we are looking to calculate the area under the curve from 𝑥𝑥 𝑡𝑡𝑡𝑡 ∞, 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑥𝑥 = 0.7.

𝑃𝑃 𝑥𝑥 > 3.25 = 𝑃𝑃 0.733 = 0.2420

We then sum the probabilities of each range to calculate the percentage the proportion of the workers getting wages between $2.75 and $3.69 an hour: 𝑃𝑃 𝑥𝑥 < 3.25 − 𝑃𝑃 𝑥𝑥 > 3.25 = 0.7881 − 0.2420 = 0.5461 = 54.61%



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Using our calculator hack, we get an answer of 0.566 or 56.6%, which has a percent

difference of 1.05% from the method of solving for the problem by hand. On the FE

Exam, we should look for the closest answer choice, and realize that NCEES will not put any choices that have a percent difference of less than 10%.

Therefore, the correct answer is 𝐁𝐁. 𝟓𝟓𝟓𝟓. %



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