Topic 2 - Measures Of Central Tendency, Variability and Linear Relationships

This article is a topic within the subject Business & Economic Statistics.

Required Reading

Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition, pp. 98-144.

Measures of Central Location/Tendency

Arithmetic Mean - Average

• This can only be used for interval data

Median

• Middle Number (of the data)
• Can be used for interval and ordinal data but not nominal

Mode

• Greatest frequency
• Can be used for all types of data

Which is Best?

• Mean is generally first selection, however median is not as sensitive to extreme values
• Mode is seldom the best

Measures of Variability

Range

• Max Value - Min Value

Variance

• Population Variance
• Sample Variance

Standard Deviation

• Square root the Variance = ‘σ’ or ‘S’

Z Score

• Z = (Observation – Mean) / Standard Deviation

Coefficient of Variation

• removes impact of differing magnitude of numbers - it is a relative measure of variability

Measures of Relative Standing And Box Plots

Percentiles

Percentiles is a ranking of data whereby P percent of observations are lower than that value.

• Median is the 50th percentile

  • INTER-QUARTILE RANGE = Q3 (75th percentile) – Q1 (25th percentile)

• Locating Percentiles:

  • Where 'n' is the number of values and 'p' is the percentile

Box Plots

• Inputs: Min, Max, 1st/2nd/3rd Quartiles
• 'Whiskers' extend outwards to the most extreme point that is not an outlier
• Outliers are considered to be the 1st Quartile MINUS OR 3rd Quartile PLUS 1.5 * Inter-quartile range)

Measures of Linear Relationships (Bivariate Analysis)

Covariance

• Sign determines +/-/no linear relationship Magnitude determines strength
• If positive (negative) numbers it’s a positive (negative) linear relationship

Coefficient of Correlation

• Covariance / SD(X)*SD(Y) - where SD(X) is the standard deviation of 'X'
• Judged in relation to:

  • -1 = perfect negative relationship
  • 0 = no linear relationship)
  • 1 = positive linear relationship

Coefficient of Determination - 'R'

• It is the Coefficient of Correlation SQUARED
• E.g. if S(x,y) = .87 than R^2 = .7588 (75.88% of the variation is explain by the other variable)

  • PLEASE NOTE: Correlation is not Causation - there can be confounding factors

    • E.g. Mobile Phones & Cancer. There are more mobile phone users in the city which would give them more exposure to radiations

The Least Squares Line

• Line -
• B0 & B1 chosen to minimize e(1,2…n) where e(1]^2 = (y1-y^)^2
• More on this later in the course within the topic of Regression

End

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References

Textbook refers to Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition,.