Dispersion measures the spread or variation of data values around a central measure such as the mean or median. Studying dispersion is important for understanding how clustered or spread out the data is, and for comparing the spreads of different data distributions. Common measures of dispersion include range, mean deviation, variance, and standard deviation. These quantify how far values lie from the central tendency on average.

1. Dispersion
Why Study Dispersion?
– A measure of location, such as the mean or the
median, only describes the center of the data. It is
valuable from that standpoint, but it does not tell us
anything about the spread of the data.
– For example, if the river ahead averaged 3 feet in
depth, would you want to wade across on foot
without additional information? Probably not. You
would want to know something about the variation in
the depth.
– A second reason for studying the dispersion in a set of
data is to compare the spread in two or more
distributions.

2. Samples of Dispersions

3. Measures of Dispersion
• Range
• Mean Deviation
• Variance and Standard
Deviation

4. EXAMPLE – Range
The number of cold drink sold at the railway station in the
Jabalpur between 4 and 7 p.m. for a sample of 5 days
last year were 20, 40, 50, 60, and 80. Determine the
mean deviation for the number of cold drink sold.
Range = Largest – Smallest value
= 80 – 20 = 60

5. EXAMPLE – Mean Deviation
The number of cold drink sold at the jabalpur railway
station between 4 and 7 p.m. for a sample of 5 days
last year were 20, 40, 50, 60, and 80. Determine the
mean deviation for the number of cold drink sold.

6. EXAMPLE – Variance and Standard
Deviation
The number of road accidents during the last five months
in Jabalpur, is 38, 26, 13, 41, and 22. What is the
population variance?

7. EXAMPLE – Sample Variance
The hourly wages
for a sample of
part-time
employees at
Home Depot
are: $12, $20,
$16, $18, and
$19. What is
the sample
variance?

8. Why square the differences?

9. Standard deviation

10. standard deviation
meaning
• It tells you how tightly your data is clustered around
the mean. When the bell curve is flattened (your data
is spread out), you have a large standard deviation —
your data is further away from the mean.
• When the bell curve is very steep, your data has a small
standard deviation — your data is tightly clustered
around the mean.
• For example, the graph on the left might represent an
abnormally high number of students getting scores
close to the average, while the graph on the right
represents more students getting scores away from the
average.

12. sport
• In any sport, there will be teams that rate highly at some things and
poorly at others.
• Chances are, the teams leading in the standings will not show such
disparity, but will perform well in most categories. The lower the
standard deviation of their ratings in each category, the more
balanced and consistent the team.
• Teams with a higher standard deviation will likely be more
unpredictable. For example, a team that is consistently bad in most
categories will have a low standard deviation.
• A team that is consistently good in most categories will also have a
low standard deviation.
• However, a team with a high standard deviation might be the type
of team that scores a lot (strong offense) but also concedes a lot
(weak defense), or, vice versa, that might have a poor offense but
compensates by being difficult to score on.

13. climate
• Consider the average daily maximum temperatures for two cities,
one inland and one on the coast.
• The range of daily maximum temperatures for cities near the coast
is smaller than for cities inland.
• While two cities may each have the same average maximum
temperature, the standard deviation of the daily maximum
temperature for the coastal city will be less than that of the inland
city as, on any particular day, the actual maximum temperature is
more likely to be further from the average maximum temperature
for the inland city than for the coastal one.
• If you were planning to move to a city with a temperate climate,
you would be interested in the outcome (think Jabalpur vs. Goa –
the average might be close but the spread would make a big
difference).

14. finance
• In finance, standard deviation is a representation of the risk associated
with a given security (stocks, bonds, property, etc.), or the risk of a
portfolio of securities (actively managed mutual funds, index mutual
funds, or ETFs).
• Risk is an important factor in determining how to efficiently manage a
portfolio of investments because it determines the variation in returns on
the asset and/or portfolio and gives investors a mathematical basis for
investment decisions (known as mean-variance optimization).
• The overall concept of risk is that as it increases, the expected return on
the asset will increase as a result of the risk premium earned – in other
words, investors should expect a higher return on an investment when
said investment carries a higher level of risk, or uncertainty of that return.
• When evaluating investments, investors should estimate both the
expected return and the uncertainty of future returns. Standard deviation
provides a quantified estimate of the uncertainty of future returns.

16. Correlation
• Correlation is a statistical technique that can show whether and how
strongly pairs of variables are related.
• For example, height and weight are related; taller people tend to be
heavier than shorter people. The relationship isn't perfect. People of the
same height vary in weight, and you can easily think of two people you
know where the shorter one is heavier than the taller one. Nonetheless,
the average weight of people 5'5'' is less than the average weight of
people 5'6'', and their average weight is less than that of people 5'7'', etc.
Correlation can tell you just how much of the variation in peoples' weights
is related to their heights.
• Although this correlation is fairly obvious your data may contain
unsuspected correlations.
• You may also suspect there are correlations, but don't know which are the
strongest.
• An intelligent correlation analysis can lead to a greater understanding of
your data.

18. Karl Pearson's correlation coefficient