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Statistics 1A: Describing Data with Numerical Measures

Introduction to Descriptive Statistics

• Graphical methods may not always be sufficient for describing data.
• Numerical measures can be created for both populations and samples.
  • Parameter: A numerical descriptive measure calculated for a population.
  • Statistic: A numerical descriptive measure calculated for a sample.

Describing Data with Numerical Measures

• Categories of Descriptive Statistical Measures:
  • Location
  • Shape
  • Spread

Central Location Statistics

• A central location statistic provides a single number that indicates the sense of the concentration of data values in a sample.

Measures of Location

Example 1.14: The Mean

• The mean, or arithmetic average, is a frequently utilized measure of the center for a set of numbers, often referred to as the sample mean.
• Definition: The sample mean of observations $x1, x2, …, xn$ is given by: $\bar{x} = \frac{1}{n} \sum</em>{i=1}^{n} x_i$
• Advantages:
  • Uses all data values.
• Disadvantages:
  • Only valid for numeric variables.
  • Distorted by outliers.
• Humorous Illustration: "My girlfriend dropped me – she said I am AVERAGE. But I think she was just being MEAN!"

Measures of Location (Median)

Example 1.15: The Median

• The median is not influenced by outliers but is only appropriate for numeric data.
• Calculating Mean and Median for different heart rates:
  • 19-year-old patients: 108, 68, 80, 83, 72.
  • 55-year-old patients: 86, 86, 92, 100, 112, 116, 136, 140.
• The median can be calculated as follows:
  • If $n$ is odd, the median is the middle value.
  • If $n$ is even, the median is the average of the two middle values.

Measures of Location - Population Mean and Median

• Generally, the population mean and median will not be identical.
• Skewness: If the population distribution is positively or negatively skewed, then:
  $mean ≠ median$
• Important considerations for making inferences based on population characteristics involve deciding which characteristic (mean or median) is more relevant.

Quartiles and Percentiles

• The median divides the data set into two equal parts.
• Quartiles: Divide the data set into four equal parts:
  • First quartile ($Q_1$): 25th percentile
  • Second quartile ($Q_2$): Median (50th percentile)
  • Third quartile ($Q_3$): 75th percentile
• Percentiles: For finer measures, percentiles divide the data into 100 parts. E.g., the 99th percentile separates the highest 1% from the bottom 99%.

The Trimmed Mean

• The trimmed mean excludes the first $k$ and last $(n-k)$ order statistics to reduce the impact of outliers.
• Robustness: Trimmed means are not unduly affected by extreme values.
• Example: Judges' scores in sports where extreme scores are discarded before calculation.

Measures of Variability

• Reporting a measure of center (mean or median) gives partial information about data sets.
• Samples can have the same central measures but different spreads.
  • Visual Representation: Dot plots may show varying extents of spread even with identical means and medians.

Descriptive Statistical Measures for Variability

• Types of Measures:
  • Variance
  • Standard Deviation
  • Interquartile Range (IQR)
  • Range
  • Quartile Deviation

Measures of Variability - The Range

• The range is the difference between the largest and smallest sample values:
  $R = x<em>{max} - x</em>{min}$
• Adequate for small data sets but not comprehensive.

Measures of Variability - The Interquartile Range

• The Interquartile Range (IQR) is defined as:
  $IQR = Q<em>3 - Q</em>1$
• Where $Q3$ is the median of the upper half and $Q1$ is the median of the lower half of the data set.

Variance and Standard Deviation

• Population Variance ($\sigma^2$) and Sample Variance ($s^2$):
  • Population variance formula:
    $\sigma^2 = \frac{1}{N} \sum<em>{i=1}^{N} (x</em>i - \mu)^2$
  • Sample variance formula:
    $s^2 = \frac{1}{n-1} \sum<em>{i=1}^{n} (x</em>i - \bar{x})^2$
• Standard Deviation is the square root of variance.

Variance with a Constant

• If $y = cx + d$, where $c$ is a constant:
  • Sample Variance of $y$:
    $s<em>y^2 = c^2 s</em>x^2$

Boxplots

• A boxplot is based on measures that remain stable in the presence of a few outliers, specifically the median and a measure of spread known as the fourth spread.
• Definitions for Boxplots:
  • Lower Fourth: Median of the smallest half.
  • Upper Fourth: Median of the largest half.
  • Fourth Spread ($fs$):
    $fs = upper hfourth - lower hfourth$

The Five-Number Summary

• A basic boxplot summarizes a data set with:
  • Minimum
  • Lower Fourth
  • Median
  • Upper Fourth
  • Maximum
• Boxplots provide a quick overview of data distribution and detect outliers.

Outliers in Boxplots

• Outlier Definition: Any observation farther than 1.5 times the fourth spread ($1.5fs$) from the closest fourth is considered an outlier.
  • Extreme Outlier: More than $3fs$ from the nearest fourth.
  • Mild Outlier: Within the range of $1.5fs$ from the nearest fourth.

Interpreting Box Plots

• Characteristics of distributions based on boxplot structure:
  • Symmetric Distribution: Median line in center of box and whiskers of equal length.
  • Skewed Right: Median line left of center and long right whisker.
  • Skewed Left: Median line right of center and long left whisker.

Comparative Boxplots

• Used effectively to reveal similarities and differences between two or more data sets concerning the same variable.
  • Example: Box plots of marks from 450 students across three classes can illustrate their comparative performance.