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Normal Distribution/Gaussian Distribution

Normal distribution is a continuous probability distribution.

Its graph is called normal curve.

 

Johann Carl Friedrich Gauss, a German mathematician, played a major role in deriving the Normal distribution, and hence, normal distribution is also known Gaussian distribution.

 

A continuous random variable x is said to follow Normal distribution if it has a probability density functionDescription: 68651.png

 

Where μ is the mean of the distribution and Description: 68664.pngis the variance.

 

A normal variate with parameters μ and σ is denoted by N(μσ).

Properties of Normal Distribution

  • It is a bell-shaped curve.
  • The distribution is symmetrical about the mean μ.
  • Since the curve is symmetric about μ, the mean, mode and median coincide, i.e.,
     
    mean = mode = median.
  • It is unimodal distribution (single mode).
  • The standard deviation for a normal distribution is given by σ.
  • The mean deviation is MD = 4/5σ. The quartile deviation is Q.D = 2/3σ
  • The quartiles are given by
    • Lower quartile Description: 68717.png
    • Upper quartile Description: 68723.png
  • So the value of quartile deviation is 0.675 σ.
  • (μ + σ) and (μ - σ) are known as the points of inflexion, i.e., the curve changes its curvature from concave to convex and form convex to concave at the two points, respectively.
  • Asymptotic base: The two tails of the curve extend indefinitely and never touch the horizontal axis.
  • The area under the curve between two limits is constant, i.e.,
    • Area between Description: 68731.png is 0.6828 of the total area under the curve
    • Area between Description: 68737.png is 0.9846 of the total area under the curve
    • Area between Description: 68743.png is 0.9973 of the total area under the curve

Applications of Normal Distribution

  • Normal distribution can be manipulated for social purposes and natural sciences.
  • Binomial distribution tends to normal distribution when the number of trials is very large and the probability of success is moderate.
  • Poisson distribution also tends to normal distribution for large values of m.
  • It is very useful in statistical quality control.

Standard Normal Distribution

Standard normal distribution is very similar to a normal distribution only in this case, the mean, median and mode are all equal to zero and the standard deviation is equal to 1.

So,Description: 68755.png

 

A continuous random variable z is said to follow normal distribution if it has a probability density functionDescription: 68761.png

 

Substituting μ = 0 and σ2 = 1, we get the probability function for standard normal distribution for a continuous random variable z as

Description: 69249.png 

Properties of Standard Normal Distribution

The probability curve for standard normal distribution can be obtained by plotting the function y = f(x).

  • As we saw earlier, mean = mode = median = 0.
  • The standard deviation is Description: 69262.png
  • The mean deviation is given by 0.8σ = 0.8.
  • The quartile deviation is given by 0.675σ = 0.675.
  • The two points of inflexion are +1 and -1.

Example
The lifetimes of certain kinds of electronic devices have a mean of 300 hours and standard deviations of 25 hours. Assuming that the distribution of these lifetimes, which are measured to the nearest hour, can be approximated closely with a normal curve,
a. find the probability that any one of these electronic devices will have a lifetime of more than 350 hours
b. find the percentage of lifetimes of these electronic devices more than 300 hours or less
c. find the percentage of lifetimes of these electronic devices will have lifetimes from 220 or 260 hours
Solution
Given μ = 300, σ = 25 and x = 350, then Description: 69277.png The area under normal curve between z = 0 and z = 2 is 0.9772.
Thus, the required probability is 1 – 0.9772 = 0.0228.
a. Description: 69290.png    ∴ The required percentage is 0.5000 x 100 = 50%.
b. Given x1 = 220, x2= 260, μ = 300 and σ = 25. Thus, Description: 69297.png
From the normal table, we have P (z = -1.6) = 0.4452 and P(z = - 3.2) = 0.4903.
Thus, the required probability is P(z = -3.2) - P(z = -1.6) = 0.4903 - 0.4452 = 0.0541.

 

Example
Assume that the test scores from a college admissions test are normally distributed and it should be in order with a mean of 450 and a standard deviation of 100.
A. What percentage of people taking the test score are between 400 and 500?
B. Suppose, someone received a score of 630. What percentage of people taking the test score better? What percentage score worse?
C. If particular university will not admit any one scoring below 480, what percentage of persons taking the test would be acceptable to that university?
Solution
A. Given μ = 450, σ = 100. Let x be the test score, then
Description: 69308.png 
The area under normal curve between z = 0 and z = 0.5 is 0.1915. Thus, the required probability that the score falls between 400 and 500 is P(400 ≤ x ≤ 500) = P(-0.5 ≤ z ≤ 0.5) = 0.1915 + 0.1915 = 0. 3830.
 
So, the percentage of the people taking the test score between 400 and 500 is 38.30%.
 
B. Given x = 630, μ = 450, σ = 100. Let x be the test score, thenDescription: 69316.png
 
The area under normal curve between z = 0 and z = 1.8 is 0.4641.
 
The probability that people taking the test score better is given by:
 
P(x ≥ 630) = P(z ≥ 1.8) = 0.5000 + 0.4641 = 0.9640, i.e., 96.40% people score better.
 
The probability that people taking the test score worse is given by
 
P(x ≥ 630) = P(z ≥ 1.8) = 0.5000 - 0.4641 = 0.0.0359.
 
So, 3.59 percent people score worse.
C. Given x = 480, μ = 450, σ = 100. Let x be the test score. Then,Description: 69327.png The area under normal curve between z = 0 and z = 0.30 is 0.1179.
 
So, P(x ≥ 480) = P(z ≥ 0.30) = 0.5000 + 0.1179 = 0.6179.
 
The people who score more than 480 and acceptable to the university are 61.79%.

 

Example
The customer accounts of a certain departmental store have an average balance of ₹ 480 and a standard deviation of ₹160. Assuming that the balance is normally distributed,
a. What is the proportion of the accounts is over ₹600?
b. What is the proportion of the accounts is between ₹240 and Description: Indian Rupee symbol.svg360?
Solution
Let the random variable x be the valance of the customer accounts. Variable x is normally distributed with μ = 480 and σ = 160. The standard normal variable is Description: 69343.png
 
a. Description: 69360.png Hence, 22.66% of the accounts have a balance in excess of ₹600.
 
b. Probability that the accounts lie between ₹400 and ₹600 is given by P(400 ≤ x ≤ 600).
Description: 69371.png 
Then, P(400 < x < 600) = P(-0.5 < x < 0.75) = 0.1915 + 0.2734 = 0.4649
 
Hence, 46.69% of the accounts have an average balance between ₹400 and ₹600.

 

Example
An aptitude test for selecting an officer in a bank was conducted on 1000 candidates. The average score was 42 and the standard deviations of the scores is 24. Assuming the normal distribution of the scores, find
a. The no. of candidates whose scores exceed 58.
b. The no. of candidates whose scores lie between 30 and 56.
Solution
A. Number of candidates whose scores exceed 58:Description: 69378.png
Area to the right of z = 0.667 under the normal curve is (0.5 – 0.2476) = 0.2524.
Number of candidates whose scores exceed 60 is 100 x 0.2524 = 252.4 or 252.
b. Number of candidates whose scores lie between 30 and 56. Standard normal variate corresponding to 30 is Description: 69386.png
Standard normal variate corresponding to 66, Description: 69395.png 
Area between z = -0.5 and z = 1 is 0.1915 + 0.3413 = 0.5328.
Number of candidates whose scores lie between 30 and 66 is
1000 x 0.5328 = 532.8 or 533.

 

Example
The annual commissions per salesperson employed by a pharmaceutical company, which manufactured cough syrup, averaged ₹40,000 with a standard deviation of ₹5000. What percent of the salespersons earn between ₹32,000 and ₹42,000?
Solution
Given μ = 40, σ = 5. Thus, P(3200 ≤ x ≤ 42000) = Description: 69404.png = P[0.40 ≤ x ≤ -1.60]
 
= 0.1554 + 0.4452 = 0.6006, i.e., 60% (approx.)
 
Description: 69411.png 





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