Lesson 3.

2 | Normal Distribution

The normal distribution (Gaussian distribution) is a continuous probability distribution that describes the
values in a data set that clusters about the mean. The graph of the Gaussian distribution is a normal
curve defined by the density function

Abraham de Moivre (1667-1754) devised the normal curve mathematically in 1733 as an approximation
to the binomial distribution. Karl Pearson (1857-1936) discovered his work in 1924. Pierre-Simon Laplace
(1749-1827) then used the normal curve to describe the distribution of errors in 1783. During 1809, Carl
Friedrich Gauss (1777-1855) utilized the normal curve in analyzing data in astronomy. In the modern
times, the normal curve also known as the bell-shaped curve plays its important role in modeling real
life problem applications involving normal data values.

The normal curve has the following characteristics that allow users to estimate the probability of
occurrence of a phenomenon represented by the value of a normally distributed factors.

a. The distribution is bell-shaped.

b. The normal curve is symmetric about the mean.

c. The mean, median and the mode values coincide.

d. The total area under the curve is 1 or 100%.

e. The area that fall within 1 standard deviation of the mean is 68%, within 2 standard deviation,
about 95%, and with 3 standard deviation is 99.7%.

The figure below shows the area specified for each region.
Example: A survey of 200 rice distribution outlets in the province found that the selling price per sack of
rice is approximately normally distributed with a mean of Php 2250 and a standard deviation of Php 25.
How many rice distribution outlets that sells

i) less than Php 2,200 per sack of rice?

ii) between Php 2,225 and Php 2,275 per sack of rice?

iii) more than Php 2,275 per sack of rice?

Solution:

a. The Php 2,200 selling price per sack of rice is 2 standard deviations below the mean. Since 47.7% of all
the data falls between the mean and 2 standard deviation below the mean,

(47.7%) (200)=(0.477)(200)=95

of the distribution outlets sells between Php 2,200 and the mean price of Php 2,250 per sack of rice.
However, half of the 200 sacks of rice have prices less than the mean price of Php 2,250. Thus, the
number of distribution outlets that sells less than Php 2,200 price per sack is 100-95=5 sacks of rice.

b. The Php 2,225 selling price per sack of rice is 1 standard deviation below the mean while the price of
Php 2,275 is at 1 standard deviation above the mean. For a normal distribution, 68% of all the data lie
within 1 standard deviation about the mean. Hence, approximately, (68%) (200)=(0.68)(200)=136
distribution outlets sells between Php 2,225 and Php 2,275 per sack of rice.

c. The selling price of Php 2,275 per sack of rice is 1 standard deviations above the mean. In a normal
distribution, 68% of all the data falls within 1 standard deviation about the mean. This implies that 32%
of the data will fall either less than 1 standard deviation below the mean or more than 1 standard
deviation above the mean. Thus, the 16% of the data that are 1 standard deviation above the mean is
 (16%)(200)=(0.16)(200)=32 distribution outlets.

The Standard Normal Distribution

The standard normal distribution is a probability distribution that has a mean of 0 and a standard
deviation of 1. Such distribution is a result of the transformation wherein the normally distributed x-
scores are transformed into z – scores through the formula:

Example: The average age of marketing supervisors is 38 years old. Assume that the data are normally
distributed. If the standard deviation is 6 years old, find the proportion of marketing supervisors whose
age are between 35 years old and 42 years old.

Solution: The standard z - scores for 35 and 42 years old x-scores are computed as

This means that 30.85% of the data lies below z=-0.50 while 74.86% of the data values fall below z=0.67.
Since the proportion of marketing managers to be determined is between 35 years old and 42 years old
or equivalently between z=-0.50 and z=0.67, then 74.86% - 30.85% = 44.01%. Thus, the proportion of
marketing managers whose age is between 35 and 42 years old is 44.01%.

Remark: The proportion of values bounded by z-scores under the standard normal curve can also be
determined using the z table depicting the area under the standard normal curve. The notion of
proportion can also be thought as percentage, probabilities or areas when dealing with problems
involving the approximately or normally distributed data values.

Explore:

1. Using the excel “NORMSDIST” function find

a. the proportion of marketing managers whose age are greater than 38 years old.
b. the percentage of marketing managers whose age are less than 40 years old.

2. Verify the results obtained in 1-a and 1-b using the z-table.

Some online resources:

https://www.google.com/search?q=normal+distribution.ppt&oq=normal+&aqs=chrome.2.69i57j69i59l3
j0l3j69i61.3905j0j7&sourceid=chrome&ie=UTF-8#

The Normal Distribution and the 68-95-99.7 Rule (5.2). Retrieved

Random Variables and Discrete Probability Distributions. Retrieved