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Chapter 3, Numerical Descriptive Measures
• Data analysis is objective– Should report the summary measures that best
meet the assumptions about the data set
• Data interpretation is subjective– Should be done in fair, neutral and clear
manner
Summary Measures
Arithmetic Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Coefficient of Variation
Range
Interquartile Range
Geometric Mean
Skewness
Central Tendency Variation Shape
Quartiles
Arithmetic Mean• The arithmetic mean (mean) is the most common measure of
central tendency
• Mean = sum of values divided by the number of values
• Affected by extreme values (outliers)
Sample size
n
XXX
n
XX n21
n
1ii
Observed values
Geometric Mean• Geometric mean
– Used to measure the rate of change of a variable over time
• Geometric mean rate of return– Measures the status of an investment over time
– Where Ri is the rate of return in time period I
n/1n21G )XXX(X
1)]R1()R1()R1[(R n/1n21G
Median: Position and Value• In an ordered array, the median is the “middle”
number (50% above, 50% below)• The location (position) of the median:
• The value of median is NOT affected by extreme values
dataorderedtheinposition2
1npositionMedian
Mode
• A measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• Used for either numerical or categorical data
• There may may be no mode
• There may be several modes
Quartiles• Quartiles split the ranked data into 4 segments
with an equal number of values per segment• Find a quartile by determining the value in the
appropriate position in the ranked data, where
First quartile position: Q1 = (n+1)/4
Second quartile position: Q2 =2 (n+1)/4 (the median position)
Third quartile position: Q3 = 3(n+1)/4
where n is the number of observed values
Same center,
different variation
Measures of VariationVariation
Variance Standard Deviation
Coefficient of Variation
Range Interquartile Range
• Measures of variation give information on the spread or variability of the data values.
Range and Interquartile Rage
• Range– Simplest measure of variation
– Difference between the largest and the smallest observations:
Range = Xlargest – Xsmallest
– Ignores the way in which data are distributed
– Sensitive to outliers
• Interquartile Range– Eliminate some high- and low-valued observations and calculate
the range from the remaining values
– Interquartile range = 3rd quartile – 1st quartile
= Q3 – Q1
• Average (approximately) of squared deviations of values from the mean
– Sample variance:
Variance
1-n
)X(XS
n
1i
2i
2
Where = arithmetic mean
n = sample size
Xi = ith value of the variable X
X
Standard Deviation• Most commonly used measure of variation
• Shows variation about the mean
• Has the same units as the original data
• It is a measure of the “average” spread around the mean
• Sample standard deviation:
1-n
)X(XS
n
1i
2i
Coefficient of Variation
• Measures relative variation
• Always in percentage (%)
• Shows variation relative to mean
• Can be used to compare two or more sets of data measured in different units
100%X
SCV
Shape of a Distribution
• Describes how data are distributed
• Measures of shape– Symmetric or skewed
Mean = Median Mean < Median Median < Mean
Right-SkewedLeft-Skewed Symmetric
Using the Five-Number Summary to Explore the Shape
• Box-and-Whisker Plot: A Graphical display of data using 5-number summary:
• The Box and central line are centered between the endpoints if data are symmetric around the median
Minimum, Q1, Median, Q3, Maximum
Min Q1 Median Q3 Max
Distribution Shape and Box-and-Whisker Plot
Right-SkewedLeft-Skewed Symmetric
Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3
• If the data distribution is bell-shaped, then the interval:– contains about 68% of the values in the population or
the sample
– contains about 95% of the values in the population or the sample
– contains about 99.7% of the values in the population or the sample
Relationship between Std. Dev. And Shape: The Empirical Rule
1σμ
2σμ
3σμ
Population Mean and Variance
N
μ)(Xσ
N
1i
2i
2
Population variance
N
XXX
N
XN21
N
1ii
Population Mean
Covariance and Coefficient of Correlation
• The sample covariance measures the strength of the linear relationship between two variables (called bivariate data)
• The sample covariance:
• Only concerned with the strength of the relationship
• No causal effect is implied
1n
)YY)(XX()Y,X(cov
n
1iii
• Covariance between two random variables:• cov(X,Y) > 0 X and Y tend to move in the same direction
• cov(X,Y) < 0 X and Y tend to move in opposite directions
• cov(X,Y) = 0 X and Y are independent
• Covariance does not say anything about the relative strength of the relationship.
• Coefficient of Correlation measures the relative strength of the linear relationship between two variables
YXn
1i
2i
n
1i
2i
n
1iii
SS
)Y,X(cov
)YY()XX(
)YY)(XX(r
• Coefficient of Correlation:
– Is unit free
– Ranges between –1 (perfect negative) and 1(perfect
positive)
– The closer to –1, the stronger the negative linear
relationship
– The closer to 1, the stronger the positive linear
relationship
– The closer to 0, the weaker any positive linear
relationship
– At 0 there is no relationship at all
Correlation vs. Regression• A scatter plot (or scatter diagram) can be used
to show the relationship between two variables
• Correlation analysis is used to measure strength of the association (linear relationship) between two variables– Correlation is only concerned with strength of the
relationship
– No causal effect is implied with correlation