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Normal Distribution Imp ERP

Normal distribution, or Gaussian distribution, is a symmetric distribution characterized by its mean (μ) and standard deviation (σ). Key properties include its bell-shaped curve, the relationship between mean, median, and mode, and the empirical rule which states that approximately 68%, 95%, and 99.7% of observations fall within 1, 2, and 3 standard deviations from the mean, respectively. The standard normal distribution is a special case with a mean of 0 and standard deviation of 1, and probabilities can be calculated using z-scores.

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11 views4 pages

Normal Distribution Imp ERP

Normal distribution, or Gaussian distribution, is a symmetric distribution characterized by its mean (μ) and standard deviation (σ). Key properties include its bell-shaped curve, the relationship between mean, median, and mode, and the empirical rule which states that approximately 68%, 95%, and 99.7% of observations fall within 1, 2, and 3 standard deviations from the mean, respectively. The standard normal distribution is a special case with a mean of 0 and standard deviation of 1, and probabilities can be calculated using z-scores.

Uploaded by

saketsahu1601
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Normal Distribution

Normal or Gaussian distribution is a symmetric distribution. Normal


distribution has two parameters: mean (μ) the location parameter and
standard deviation (σ), the spread parameter.
If a X is a continuous random variable with mean μ and standard deviation σ,
then it is written as, X ~ N(μ , σ). Note that, the bars in the histogram below
must be connected to each other, not separate.

Source: emeraldinsight.com

Properties of Normal distribution

1. This is a distribution for continuous random variable. Therefore


probability is computed by measuring the area under the curve rather
than the curve height or frequency or count.

2. Normal distribution curve is bell-shaped, symmetric around its mean.

3. The normal distribution has a single mode. That is why this is also known
as a unimodal distribution.

4. For normal distribution, mean = median = mode.


5. The probability distribution is highest exactly at the mean (in other
words, mean has the highest frequency, it is the most likely value for the
random variable).

6. Most of the statistical inference theory is based on the properties of the


normal distribution.

7. Total area under the normal distribution curve is 1.

The Empirical Rule

Source: pdnotebook.com

If the histogram of values in a data set can be reasonably approximated by a


normal curve, then

a) approximately 68% of the observations are within 1 standard


deviation of the mean;
b) approximately 95% of the observations are within 2 standard
deviations of the mean;
c) approximately 99.7% of the observations are within 3 standard
deviations of the mean.
Normal distribution has two parameters – mean (μ) the location parameter
and standard deviation (σ), the spread parameter.

psychology.wikia.com

Changing the mean will cause the distribution to shift its location. Standard
deviation controls the spread of the distribution; the larger it is the wider is
the distribution shape.

Standard normal distribution

This is a normal distribution with mean μ = 0 and standard deviation σ = 1.

If a variable x has any normal distribution with mean μ and standard


deviation σ, then the standardized variable

z = (x – μ)/σ

has the standard normal distribution, z ~ N(0,1).


Finding normal probabilities

Areas under a normal curve represent proportions of observations from that


normal distribution. So they represent probabilities of randomly selecting an
individual from that distribution.

Cumulative areas or probabilities under the standard normal curve are


available in a table form. Standard normal curve is symmetric, the
distribution can be divided into two equal parts at μ = 0. So all the negative z
values are on the left side and positive z values are on the right side of the
curve. Probabilities on the left side of the curve are the same as the
probabilities on the right side of the curve.

The shaded region in this


graph represents the
cumulative probability that z is
less than or equal to a, that is
P(z ≤ a).

Section 6.1

The random variable x has normal distribution with mean μ and variance σ2.
We can find probabilities for three different cases:

Case 1: P(x ≤ a). Find Z score for ‘a’ and then use the z table.

Case 2: P(x ≥ b) = 1 – P(x ≤ b). Find Z score for ‘b’ and then use the z
table.
Case 3: P(a ≤ x ≤ b) = P(x ≤ b) – P(x ≤ a). Find Z scores for both ‘a’ and
‘b’ and then use the z table.

The z score formula is z = (x – μ)/σ.

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