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Probability: Normal Distribution

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The normal distribution refers to a family of continuous probability distributions described by the normal equation.

Normal Distribution

on the domain x ∈ (- ∞, ∞)

where x is a normal random variable, μ is the mean, σ is the standard deviation,

Thus the normal distribution can be completely specified by two parameter mean (μ) and standard deviation (σ) and is represented as N(μ, σ).

Normal Distribution

Mathematicians called this distribution a normal distribution, a physicist called it a Gaussian distribution, and scientists called it a bell curve due to its bell-like shape.

The normal distribution with mean μ = 0 and standard deviation, σ = 1 is called the standard normal distribution. It is denoted by N(0, 1).

Characteristics of a Normal Distribution

  • The normal curve is symmetrical about the mean μ. It is perfectly symmetrical around its center. That is, the right side of the center is a mirror image of the left side.
  • The mean is at the middle and divides the area into halves. The center of a normal distribution is located at its peak, and 50% of the data lies above the mean, while 50% lies below. It means that the mean, median, and mode are all equal in a normal distribution.
  • There is also only one mode, or peak, in a normal distribution.
  • Normal distributions are continuous and have tails that are asymptotic.
  • The total area under the curve is equal to 1;
  • It is completely determined by its mean and standard deviation (SD) σ (or variance σ2)
  • Approximately 68 % of the data lies within 1 SD of the mean. Approximately 95 % of the data lies within 2 SD of the mean.  Approximately 99.7 % of the data lies within 3 SD of the mean.

The Z – score:

The number of standard deviations from the mean is called the standard score or z – score.

An arbitrary normal distribution can be converted to a standard normal distribution by changing variables to z.

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Empirical Rules for z – Scores:

  • Approximately 68 % of the data lies within 1 SD of the mean. 
  • Approximately 95 % of the data lies within 2 SD of the mean. 
  • Approximately 99.7 % of the data lies within 3 SD of the mean.

Importance of z – Score:

Z-Scores tell us whether a particular score is equal to the mean, below the mean or above the mean of a bunch of scores. They can also tell us how far a particular score is away from the mean.

Z-ScoreConclusion
0It is equal to the group mean
PositiveIt is above the group mean
NegativeIt is below the group mean
+ 1It is 1 Standard Deviation above the mean
+ 2It is 2 Standard Deviation above the mean
– 1It is 1 Standard Deviation below the mean
– 2It is 1 Standard Deviation below the mean

We can use Z-scores to standardize scores from different groups of data. Then we can compare raw scores from different groups of data.

Example – 01:

95 % of students at the college are between 1.1 m and 1.7 m tall. Find mean and the standard deviation assuming a normal distribution.

Solution:

Mean = μ = average of the two values given = (1.1 + 1.7)/2 = 2.8/2 = 1.4 m

From empirical rule states that approximately 95 % of the data lies within 2 SD of the mean on either side.

Therefore the total deviation is 4 S.D.

4 S.D. = 1.7 – 1.1

4 S.D. = 0.6

1 S.D. = σ = 0.15 m

Ans: Mean = 1.4 m and the standard deviation is 0.15 m

Example – 02:

95 % of students in a class of 100 weigh between 62 kg and 90 kg. Find mean and standard deviation assuming a normal distribution.

Solution:

Mean = μ = average of the two values given = (62 + 90)/2 = 152/2 = 76 kg

From empirical rule states that approximately 95 % of the data lies within 2 SD of the mean on either side.

Therefore the total deviation is 4 S.D.

4 S.D. = 90 – 62

4 S.D. = 28

1 S.D. = σ = 7 kg

Ans: Mean = 76 kg and standard deviation is 7 kg

Example – 03:

68 % of marks of students in a certain test are between 51 and 64. Find mean and standard deviation assuming a normal distribution.

Solution:

Mean = μ = average of the two values given = (51 + 64)/2 = 115/2 = 57.5 kg

From empirical rule states that approximately 68 % of the data lies within 1 SD of the mean on either side.

Therefore the total deviation is 2 S.D.

2 S.D. = 64 – 51

2 S.D. = 13

1 S.D. = σ = 6.5 kg

Ans: Mean marks = 57.5 and standard deviation in marks is 6.5

Example – 04:

99.7 % of electrical components produced by a machine have lengths between 1.176 cm and 1.224 cm. Find mean and standard deviation assuming a normal distribution.

Solution:

Mean = μ = average of the two values given = (1.176 + 1.224)/2 = 2.4/2 = 1. 2 cm

From empirical rule states that approximately 99.7 % of the data lies within 3 SD of the mean on either side.

Therefore the total deviation is 6 S.D.

6 S.D. = 1.224 – 1.176

6 S.D. = 0.048

1 S.D. = σ = 0.008 cm

Ans: Mean length = 1.2 cm and standard deviation in length is 0.008 cm

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