Descriptive Statistics Notes

Descriptive Statistics Overview

• Descriptive Statistics: A branch of statistics that provides summaries about the sample and measures.

Key Terms to Understand

• Central Tendency: Indicates the average or center of a data set.
• Variation: Describes how much the data varies or spreads out.
• Skewness: Measures asymmetry of the probability distribution of a real-valued random variable.
• Kurtosis: Measures the tailedness of the probability distribution.

Learning Objectives

Know

• Definitions of central tendency, variation, skewness, and kurtosis.
• Three principal measures of central tendency (mean, median, mode) and their application.
• Types of mean (arithmetic, weighted, geometric, harmonic).

Understand

• Importance of calculating summary statistics.
• Computation of the following measurements:
  • Arithmetic mean
  • Weighted mean
  • Geometric mean
  • Harmonic mean
  • Range
  • Mean deviation
  • Variance
  • Standard deviation
  • Coefficient of Variation
• Impact of skewness on the relative positions of the three measures of central tendency.

Be Able to

• Compute various measures of central tendency and variation.
• Create frequency distributions and histograms.
• Assess skewness and kurtosis within data distributions.

Measures of Central Tendency

Mean

• Arithmetic Mean: Sum of all observations divided by the number of observations.
  Formula: ar{Y} = rac{1}{n} imes extstyle{igg( extstyle{igg( extstyle{igg( Y1 + Y2 + ar{Y} + … + Y_n} igg) } igg) } igg)}

  • Excel Command: =AVERAGE(A1:An)

• Weighted Mean: The mean calculated by giving different weights to different observations.
  Formula: $\bar{Y} = \frac{ extstyle{\bigg( extstyle{\bigg( extstyle{\bigg( w<em>1Y</em>1 + w<em>2Y</em>2 + … + w<em>nY</em>n} \bigg) }\bigg) }\bigg)}{ extstyle{\bigg( w<em>1 + w</em>2 + … + w_n} \bigg)}$

• Geometric Mean: Use when values have multiplicative relationships.
  Formula: $GM = ext{n-th root of } extstyle{\bigg(Y<em>1 imes Y</em>2 imes … imes Y_n} \bigg)$

  • Excel Command: =GEOMEAN(A1:An)

• Harmonic Mean: Useful for rates. Formula: $H = \frac{n}{ extstyle{\bigg( \frac{1}{Y<em>1} + \frac{1}{Y</em>2} + … + \frac{1}{Y_n} \bigg)} }$

  • Excel Command: =HARMEAN(A1:An)

• Median: The middle value when data is arranged in order.

  • Excel Command: =MEDIAN(A1:An)

• Mode: The most frequently occurring value in a distribution.

  • Excel Command: =MODE(A1:An)

Relationship Between Shape of Distribution and Averages

• In a symmetric distribution:
  • Mean = Median = Mode
• In Negatively Skewed Distribution:
  • Mean < Median < Mode
• In Positively Skewed Distribution:
  • Mean > Median > Mode

Measures of Dispersion

• Range: The difference between the highest and lowest value.
  Formula: $ext{Range} = ext{Max} - ext{Min}$
• Interquartile Range (IQR): Difference between the 3rd quartile (Q3) and 1st quartile (Q1).
  Formula: $ext{IQR} = Q3 - Q1$
• Mean Deviation: The average of the absolute deviations from the mean.
  Formula: $MD = \frac{1}{n} imes extstyle{\bigg( |Y<em>1 - \bar{Y}| + |Y</em>2 - \bar{Y}| + … + |Y_n - \bar{Y}| \bigg) }$
• Variance: The average squared deviation from the mean. Formula: $V = \frac{1}{n} imes extstyle{\bigg( (Y<em>1 - \bar{Y})^2 + (Y</em>2 - \bar{Y})^2 + … + (Y_n - \bar{Y})^2 \bigg) }$
  • Excel Command: =VAR(A1:An)
• Standard Deviation: The square root of variance. Formula: $SD = ext{sqrt}(V)$
  • Excel Command: =STDEV(A1:An)
• Coefficient of Variation (CV): Ratio of the standard deviation to the mean.
  Formula: $CV = \frac{SD}{\bar{Y}} imes 100$

Assessing Shape of Distribution

• Skewness and Kurtosis: Measure of asymmetry and peakedness of the distribution.
• Skewness range: -∞ to +∞.
  • Positive: Right-skewed
  • Negative: Left-skewed
  • Zero: Symmetric
• Kurtosis range: 1 to ∞.
  • Leptokurtic: Peaked distribution
  • Platykurtic: Flat distribution
  • Mesokurtic: Normal distribution (Kurtosis = 3)