Numerical Descriptive Measures

Numerical Descriptive Measures

• The chapter covers measures of central tendency, variation and shape in numerical data.
• It describes how to calculate descriptive summary measures for a population.
• It explains how to construct and interpret a boxplot.
• It outlines how to calculate covariance and the coefficient of correlation.

Summary Definitions

• Central tendency: The extent to which data values cluster around a typical or central value.
• Variation: The amount of dispersion or scattering of values.
• Shape: The distribution pattern of values from the lowest to the highest.

Measures of Central Tendency: The Mean

• Arithmetic mean (often just called "mean"): The most common measure, calculated by summing all values and dividing by the number of values.
• For a sample of size 'n':
  • x̄ = Σxᵢ / n
  • x̄ is pronounced x-bar
  • xᵢ represents the ith value
  • n is the sample size

Example

• Employee ages (sample of 8): 53, 32, 61, 27, 39, 44, 49, 57
• Mean age: 45.25 years

Mean for Grouped Data

• For grouped data, mean is calculated as:
  • x̄ = Σ(mf) / Σf
  • 'm' is the midpoint of the class
  • 'f' is the frequency of the class

Example

• Number of orders received daily in 50 days:
  • 10-12: 4 days
  • 13-15: 12 days
  • 16-18: 20 days
  • 19-21: 14 days
• Mean: 16.64 orders

Measures of Central Tendency : The Median

• Median: The middle value in an ordered array (smallest to largest).
• If 'n' (number of values) is odd, the median is the middle number.
• If 'n' is even, the median is the average of the two middle numbers.
• Not affected by extreme values.

Measures of Central Tendency : Locating the Median

• Median position = (n+1)/2.

Measures of Central Tendency : The Mode

• Mode: The value that occurs most often in a data set.
• Not affected by extreme values.
• Can be used for numerical or categorical data.
• There may be multiple modes (multimodal) or no mode at all.

Measures of Variation: The Range

• Range: The difference between the largest and smallest values.
• Range = Xlargest - Xsmallest

Measures of Variation: Why the Range Can Be Misleading

• Ignores the way data is distributed.
• Sensitive to outliers (extreme values)

Measures of Variation: The Variance

• Variance: Average (approximately) of squared deviations of values from the mean.
• Sample Variance (S²) :
• S² = Σ(xᵢ - x̄)² / (n-1)
• Where:
  • x̄ is the arithmetic mean
  • n is the sample size
  • xᵢ is the ith value of the variable X

Measures of Variation: The Standard Deviation

• Standard Deviation: The square root of the variance.
• Has the same units as the original data.
• Typically calculated to understand the variability of the sample around the mean
• Sample Standard Deviation (S) :
• S = √Σ(xᵢ - x̄)² / (n-1)

Short Cut Formula:

• S = √(Σx² - (Σx)²/n) / (n -1)

Calculating The Interquartile Range (IQR)

• The difference between the third quartile (Q3) and the first quartile (Q1).
• IQR = Q₃ - Q₁

The Five-Number Summary

• Xsmallest
• First Quartile (Q₁)
• Median (Q2)
• Third Quartile (Q3)
• Xlargest

The Boxplot:

• A graphical display of data based on the five-number summary, showing spread and skewness.

The Covariance

• Measures the strength of the linear relationship between two numerical variables (X and Y).
• Sample Covariance (Cov(X,Y)):
• Cov(X,Y) = Σ(xᵢ -X̄)(yᵢ - ȳ) / (n-1)

Coefficient of Correlation

• A unit-free measure of the linear relationship between two numerical variables (X and Y).
• Sample correlation coefficient (r):
• r = cov(X,Y) / Sₓ * Sᵧ

Features of the Coefficient of Correlation

• Always between -1 and 1.
• Closer to +1 : strong positive linear relationship.
• Closer to -1 : strong negative linear relationship.
• Closer to 0 : weak linear relationship.

Pitfalls in Numerical Descriptive Measures

• Data analysis is objective.
• Report only necessary summary measures to describe the data set.
• Data interpretation is subjective.
• Interpretations must be fair and objective.

Ethical Considerations

• Document both good and bad results in data analysis.
• Present data in a fair and objective manner.
• Avoid distorting facts through data manipulation

Chapter Summary

• Covers various numerical descriptive statistical tools and applications

This quiz covers essential concepts of numerical descriptive measures, including measures of central tendency, variation, and shape in data. You'll learn to calculate and interpret various descriptive statistics, including the mean and boxplots. Test your understanding of these foundational statistical principles.