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