Describing Data Using Numerical Measures

Topics

Summary Measures

Measures of Central Tendency

Measures of Central Tendency

1 2

1

. .....

nn

n

ii

G x x x

x

1 2

1 1 1...

n

nH

x x x

Mean (Arithmetic Average)

Mean (Arithmetic Average)

Median

Median

Median Example

Mode

Weighted Mean

Geometric Mean The geometric mean indicates the central tendency or typical

value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root (where n is the count of numbers) of the product of the numbers.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is 2√2 × 8 = 4.

1 21

. .....n

nn i

i

G x x x x

The geometric mean only applies to positive numbers in order to avoid taking the root of a negative product

In statistical surveys when proportional differences are more important than the absolute differences, geometric mean referred instead of arithmetic mean.

Geometric Mean

Harmonic Mean

The harmonic mean H is defined to be the reciprocal of the arithmetic mean of the reciprocals of :

When prices are expressed in quantities (so many units per dollars) harmonic mean should be calculated.

1 2, ,..., nx x x

1 2

1 1 1...

n

nH

x x x

Shape of a Distribution

Which Measure of Central Tendency is the “best”?

Measures of Location (Measures of Statistical Dispersion)

Percentiles

Quartiles

Quartiles

Box and Whisker Plot

Constructing the Box and Whisker Plot

Shape of Box and Whisker Plots

Distribution Shape and Box and Whisker Plot

Measures of Statistical Dispersion (Variation)

Statistical Dispersion (variation)

Measures of statistical dispersion or variation give information on the spread or variability of the data values.

Range

Disadvantages of the Range

Interquartile Range

Interquartile Range Example

Variance

Degrees of Freedom (df)

Standart Deviation

Calculation Example: Sample Standart Deviation

Comparing Standart Deviations

Coefficient of Variation

Comparing Coefficients of Variation

Standardized Data Values

Standardized Population Values

Standardized Sample Values

Standardized Value Example

Using Probability and Probability Distributions

Important Terms

Sample Space

Events

Visualizing Events

Experimental Outcomes

Probability Concepts

Probability Concepts

Independent vs. Dependent Events

Assigning Probability

Rules of Probability

Addition Rule for Elementary Events

Complement Rule

Addition Rule for Two Events

Addition Rule Example

Addition Rule for Mutually Exclusive Events

Conditional Probability

Conditional Probability Example

Conditional Probability Example

Conditional Probability Example

For Independent Events

Multiplication Rules

Tree Diagram Example

Bayes’ Theorem

Bayes’ Theorem Example

Bayes’ Theorem Example

Bayes’ Theorem Example