This document discusses the normal distribution curve, also called the bell curve or normal curve. It describes several key properties of the normal distribution including that it is symmetrical around the mean, the area under the curve sums to 1, and most values cluster around the mean. The normal distribution is important because many natural phenomena and psychological variables follow this pattern. Statistical tests often assume a normal distribution of data, and the empirical rule can be used to determine what percentage of values fall within a given number of standard deviations from the mean for a normal distribution. The document provides guidance on checking if a dataset follows a normal distribution.

1. Presented by
Dr. R. RAJA, M.E., Ph.D.,
Assistant Professor, Department of EEE,
Muthayammal Engineering College, (Autonomous)
Namakkal (Dt), Rasipuram – 637408
Normal Curve in Total Quality Management
MUTHAYAMMAL ENGINEERING COLLEGE
(An Autonomous Institution)
(Approved by AICTE, New Delhi, Accredited by NAAC, NBA & Affiliated to Anna University),
Rasipuram - 637 408, Namakkal Dist., Tamil Nadu.

2. Normal Curve
Normal Curve or Normal Distribution (Bell Curve)
What are the properties of the normal distribution?
 The normal distribution is a continuous probability distribution that is symmetrical on
both sides of the mean, so the right side of the center is a mirror image of the left side.
 The area under the normal distribution curve represents probability and the total area
under the curve sums to one.
 Most of the continuous data values in a normal distribution tend to cluster around the
mean, and the further a value is from the mean, the less likely it is to occur. The tails
are asymptotic, which means that they approach but never quite meet the horizon (i.e.
x-axis).
 For a perfectly normal distribution the mean, median and mode will be the same value,
visually represented by the peak of the curve.
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 The normal distribution is often called the bell curve because the graph of its
probability density looks like a bell. It is also known as called Gaussian distribution,
after the German mathematician Carl Gauss who first described it.

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Why is the normal distribution important?
The bell-shaped curve is a common feature of nature and psychology
 The normal distribution is the most important probability distribution in
statistics because many continuous data in nature and psychology displays this
bell-shaped curve when compiled and graphed.
 For example, if we randomly sampled 100 individuals we would expect to see a
normal distribution frequency curve for many continuous variables, such as IQ,
height, weight and blood pressure.
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Parametric significance tests require a normal distribution of the samples' data
points
 The most powerful (parametric) statistical tests used by psychologists require
data to be normally distributed. If the data does not resemble a bell curve
researchers may have to use a less powerful type of statistical test, called non-
parametric statistics.
Converting the raw scores of a normal distribution to z-scores
 We can standardized the values (raw scores) of a normal distribution by
converting them into z-scores.
 This procedure allows researchers to determine the proportion of the values that
fall within a specified number of standard deviations from the mean (i.e.
calculate the empirical rule).
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Probability and the normal curve: What is the empirical rule formula?
 The empirical rule in statistics allows researchers to determine the proportion of
values that fall within certain distances from the mean. The empirical rule is
often referred to as the three-sigma rule or the 68-95-99.7 rule.
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 If the data values in a normal distribution are converted to z-scores in a standard
normal distribution the empirical rule describes the percentage of the data that
fall within specific numbers of standard deviations (σ) from the mean (μ) for
bell-shaped curves.
 The empirical rule allows researchers to calculate the probability of randomly
obtaining a score from a normal distribution.
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 68% of data falls within the first standard deviation from the mean. This means
there is a 68% probability of randomly selecting a score between -1 and +1
standard deviations from the mean.

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 95% of the values fall within two standard deviations from the mean. This means
there is a 95% probability of randomly selecting a score between -2 and +2
standard deviations from the mean.

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 99.7% of data will fall within three standard deviations from the mean. This
means there is a 99.7% probability of randomly selecting a score between -3
and +3 standard deviations from the mean.
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How can I check if my data follows a normal distribution?
 Statistical software (such as SPSS) can be used to check if your dataset is
normally distributed by calculating the three measures of central tendency.
 If the mean, median and mode are very similar values there is a good chance
that the data follows a bell-shaped distribution (SPSS command here).
 It is also advisable to a frequency graph too, so you can check the visual shape
of your data (If your chart is a histogram, you can add a distribution curve using
SPSS: From the menus choose: Elements > Show Distribution Curve).
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 Normal distributions become more apparent (i.e. perfect) the finer the level of
measurement and the larger the sample from a population.
 You can also calculate coefficients which tell us about the size of the
distribution tails in relation to the bump in the middle of the bell curve. For
example, Kolmogorov Smirnov and Shapiro Wilk tests can be calculated using
SPSS.
 These tests compare your data to a normal distribution and provide a p-value,
which if significant (p < .05) indicates your data is different to a normal
distribution (thus, on this occasion we do not want a significant result and need
a p-value higher than 0.05).
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15. Thank You
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