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Numerical Descriptive
Techniques
Statistics for Management
and Economics
Chapter 4
Objectives
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Measures of Central Location
Measures of Variability
Measures of Relative Standing and
Box Plots
Measures of Linear Relationship
Graphical vs. Numerical Techniques
Data Exploration
Measures of Central Location
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Numerical measure of the center, or
middle, of the data
Arithmetic Mean
Median
Mode
Geometric Mean
Measure of Center:
Arithmetic Mean
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The arithmetic mean, a.k.a. average, shortened to
mean, is the most popular & useful measure of central
location.
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Appropriate for describing interval data.
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The arithmetic mean for a population is denoted with
Greek letter “mu”: 
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The arithmetic mean for a sample is denoted with an
“x-bar”:
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It is computed by simply adding up all the
observations and dividing by the total number of
observations:
Sum of the observations
Mean =
Number of observations
Arithmetic Mean
Population Mean
Sample Mean
Measure of Center: Median
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The median is another useful measure of central
location.
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Appropriate for describing interval or ordinal data.
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Best measure of central location when dealing with
data that has extreme values.
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Computed the same for population and sample.
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Calculated by placing all the observations in order;
the observation that falls in the middle is the
median.
HINT!
The middle of the dataset falls
at the location (n+1)/2
Measures of Center: Mode
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The mode of a set of observations is the value
that occurs most frequently.
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A set of data may have one mode (or modal
class), or two, or more modes.
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Mode is a useful measure of center for all data
types, though mainly used to identify the group
with the highest frequency for nominal data.
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For large data sets the modal class is much
more relevant than a single-value mode.
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Computed the same for population and sample.
Mean vs. Median vs. Mode
Mean and median for a
symmetric distribution
Mean
Median
Mode
Left skew or
Negative skew
Mean
Median
Mode
Mean and median for
skewed distributions
Mean
Median
Mode
Right skew or
Positive skew
Mean vs. Median vs. Mode
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Symmetric distribution: the mean, median, and
mode will be approximately the same.
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Multimodal distribution: report the mean, median
and/or mode for each subgroup.
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Nominal data: Mode calculation is useful for
determining highest frequency but not “central
location”; the calculation of the mean is not valid.
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Ordinal data: Median is appropriate; the calculation
of the mean is not valid.
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Interval data: Mean is appropriate; in the case of
skewed data, report the median as well.
Measures of Center:
Geometric Mean
The geometric mean is used when the variable is a
growth rate or rate of change, such as the value of an
investment over periods of time.
If Ri denotes the rate of return in period i (i = 1, 2, …, n), then
The geometric mean Rg of the returns R1, R2, … Rn is defined
such that:
Measures of Center:
Geometric Mean
Solving for Rg we produce the following formula:
The upper case Greek Letter “Pi” represents a product of terms…
Measures of Center:
Summary
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Use the…
Mean
To describe…
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The central location of
a single set of interval
data
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Median
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Mode
the central location of
a single set of interval
or ordinal data
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a single set of nominal
data
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Geometric Mean
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a single set of interval
data based on growth
rates
Measures of Variability
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Tell how variable, or spread out, the data
falls around the mean
Used in conjunction with measures of center
to describe a distribution with numbers
Used primarily for interval data
Three measures:
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Range
Variance
Standard Deviation
Measures of Variability:
Range
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Simplest measure of variability, easily
computed
Calculated as:
largest observation – smallest observation
Not very descriptive of the variability of
the data – how?
Measures of Variability:
Variance
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Widely used
Used to summarize data but also plays an
important role in statistical inference
In general, explains how data is spread about
the mean.
For the population, denoted by the lower
case Greek letter sigma (squared): 2
For the sample, s2
Measures of Variability:
Variance
population mean
The variance of a
population is:
population size
sample mean
The variance of a
sample is:
Note! the denominator is sample size (n) minus one !
Shortcut: Calculating Variance
A short-cut formulation to calculate sample variance
directly from the data without the intermediate step of
calculating the mean…
Measures of Variability:
Standard Deviation
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Square root of the variance
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Population:
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Sample:
Interpretation:
Standard Deviation
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Together with the sample mean, the
standard deviation can be used to “build”
the picture of a distribution.
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It can also be used to compare the
variability of different distributions. To do
this, we can use…
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The Empirical Rule
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Chebysheff’s Theorem
Interpretation:
The Empirical Rule
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For distributions with bell shaped
histograms.
States that…
1)
2)
3)
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Approximately 68% of all observations fall
within one standard deviation of the mean.
Approximately 95% of all observations fall
within two standard deviations of the mean.
Approximately 99.7% of all observations fall
within three standard deviations of the mean.
A.K.A. The 68% - 95% - 99.7% Rule
Interpretation:
Chebysheff’s Theorem
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Applies to all shapes of histograms (not
limited to bell shaped)
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The proportion of observations in any
sample that lie within k standard deviations
of the mean is at least:
Note: The Empirical Rule provides approximate proportions given
the limits where Chebysheff provides the lower bound on the
proportions.
Measures of Variability:
Coefficient of Variation
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The coefficient of variation of a set of observations
is the standard deviation of the observations divided
by their mean, that is:
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Population coefficient of variation = CV =
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Sample coefficient of variation = cv =
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This coefficient provides a proportionate measure of
variation (thus is useful for comparing variation
among two datasets).
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For example, a standard deviation of 10 may be perceived
as large when the mean value is 100, but only moderately
large when the mean value is 500.
Measures of Relative Standing
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Measures of relative standing are designed to
provide information about the position of particular
values relative to the entire data set.
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Percentile: the Pth percentile is the value for which
P percent are less than that value and (100-P)% are
greater than that value.
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Specifically, the 25%, 50%, and 75% percentiles
are Quartiles.
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You may also see fifths – Quintiles or tenths –
Deciles.
Percentiles and Quartiles…
First (lower) decile
= 10th percentile
First (lower) quartile, Q1
= 25th percentile
Second (middle) quartile,Q2
= 50th percentile
Third quartile, Q3
= 75th percentile
Ninth (upper) decile
= 90th percentile
Locating Percentiles
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The following formula allows us to
approximate the location of any
percentile:
Measures of Variability:
Interquartile Range
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Interquartile Range (IQR) = Q3 – Q1
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The interquartile range measures the
spread of the middle 50% of the
observations.
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Large values of this statistic mean that the
1st and 3rd quartiles are far apart indicating a
high level of variability.
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Usually reported with the Median (M)
Graphical Description of the
Quartiles: The Boxplot
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Sometimes also called a box-and-whisker plot
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Uses the Five Number Summary:
Minimum Q1 M Q3 Maximum
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The “box” shows the center of the data and the
general shape, the “whiskers” show the spread
of the data
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If the data extends beyond the whiskers of the
plot, there are outliers in the dataset, therefore,
this is a good summary of data with outliers!
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You can easily create side-by-side boxplots to
compare multiple groups
The Boxplot
1.5(Q3 – Q1)
1.5(Q3 – Q1)
Whisker
S
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Whisker
Q1
Q2
Q3
L
Whiskers are calculated as 1.5(Q3-Q1). In the plot
above, there is an outlier at the largest value (L)
Boxplots mimic the general shape of the
distribution.
Measures of Linear Relationship
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Numerical measures of linear relationship
that provide information as to the strength
& direction of a linear relationship (if any)
between two variables.
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Covariance - is there any pattern to the
way two variables move together?
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Coefficient of correlation - how strong is
the linear relationship between two
variables?
Measures of Linear
Relationship: Covariance
population mean of variable X, variable Y
sample mean of variable X, variable Y
Note: divisor is n-1, not n as you may expect.
Measures of Linear
Relationship: Covariance
There is also a shortcut for calculating
sample covariance directly from the
data:
Interpretation: Covariance
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When two variables move in the same
direction (both increase or both decrease),
the covariance will be a large positive
number.
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When two variables move in opposite
directions, the covariance is a large
negative number.
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When there is no particular pattern, the
covariance is a small number.
Measures of Linear
Relationship: Correlation
The Coefficient of Correlation (a.k.a., the correlation)
is the covariance divided by the standard deviations
of the variables
Greek letter
“rho”
From the correlation, we can determine the strength, direction, and linearity of
the association between X and Y. The correlation is the “numerical scatterplot”
Interpretation: Correlation
The advantage of the coefficient of correlation over covariance
is that it has fixed range from -1 to +1, thus:
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If the two variables are very strongly
positively related, the coefficient value is
close to +1 (strong positive linear
relationship).
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If the two variables are very strongly
negatively related, the coefficient value
is close to -1 (strong negative linear
relationship).
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No straight line relationship is indicated
by a coefficient close to zero.
Measures of Linear Relationship:
Least Squares Method
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Recall, the slope-intercept equation for a
line is expressed in these terms:
y = mx + b
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Where:
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m is the slope of the line
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b is the y-intercept.
If we’ve determined that a linear relationship
exists, can we determine a linear function?
Measures of Linear Relationship:
Least Squares Method
…produces a straight line drawn through the points so
that the sum of squared deviations between the points
and the line is minimized. This line is represented by the
equation:
Value of x
data (usually
given)
Estimated value of
y determined by
the line
y-intercept
slope