1.3

Introduction to Statistics

• Begin with the visual examination of data before computing statistics.
• Definition of a statistic: A numerical summary of data.
• Types of Statistics:
  • Two or more numerical summaries.
  • The discipline of displaying and interpreting data.

Describing Distributions

Measuring the Centre: Mean

• Definition: The mean (or average) is calculated as:
  $\bar{x} = \frac{1}{n} \sum<em>{i=1}^{n} x</em>i$
  where (n) is the number of observations and (x_i) represents each observation.

Example of Mean

• Given the sample ({4, 7, 6, 11, 8}):
  • Find the mean.
  • Calculation:
    $\text{Mean} = \frac{4 + 7 + 6 + 11 + 8}{5} = 7.2$

Measuring the Centre: Median

• Definition: The median (M) is the midpoint of a distribution where half the observations are smaller and the other half are larger.

Finding the Median

1. Order the Observations: Arrange all observations from smallest to largest.
2. Determine Location based on Observer Count:
   • If (n) (number of observations) is odd, (M) is the central observation: location is (\frac{n + 1}{2}) from the bottom of the list.
   • If (n) is even, (M) is the mean of the two central observations; locate the median at (\frac{n + 1}{2}) from the bottom.

Example of Median Calculation

• Given the sample ({4, 7, 6, 11, 8}):

  • Ordered: ({4, 6, 7, 8, 11})
  • Median (M = 7) (mean is 7.2).

• Given a new sample ({4, 7, 6, 11, 8, 54}):

  • Ordered: ({4, 6, 7, 8, 11, 54})
  • The median is calculated as:
    $M = \frac{7 + 8}{2} = 7.5$

Mean vs. Median

• Observation:
  • The mean is affected by outliers, meaning it is not a resistant measurement of the center of a distribution.
  • The median remains robust against outliers.

Measuring Spread: Quartiles

• Percentiles: The pth percentile is the value such that p percent of the observations fall at or below it.
• Common Quartiles:
  • First quartile (Q1) = 25th percentile.
  • Second quartile = 50th percentile (median), denoted as (M).
  • Third quartile (Q3) = 75th percentile.

Finding Quartiles

1. Order Observations: Arrange them in increasing order and locate the median (M).
2. Determine Q1: It is the median of the observations below (M).
3. Determine Q3: It is the median of the observations above (M).

Example for Quartile Calculation

• Given data: ({2, 3, 4, 6, 8, 10, 14, 17, 18, 42})
  • The five-number summary includes:
  • Minimum = 2
  • Q1 = 4
  • Median (M = 9)
  • Q3 = 17
  • Maximum = 42

The Five-Number Summary

• The five-number summary consists of:
  • Minimum
  • (Q1)
  • Median ((M))
  • (Q3)
  • Maximum

Example of the Five-Number Summary

• From data: ({2, 3, 4, 6, 8, 10, 14, 17, 18, 42}):
  • The five-number summary is ({2, 4, 9, 17, 42}).

The Interquartile Range (IQR)

• Definition: The interquartile range (IQR) is defined as:
  $IQR = Q3 - Q1$
• Example with (Q1 = 4) and (Q3 = 17):
  • Calculation:
    $IQR = 17 - 4 = 13$

The 1.5 × IQR Rule for Outliers

• An observation is suspected as an outlier if it:
  • Falls more than 1.5 × IQR above Q3, or
  • Falls more than 1.5 × IQR below Q1.

Example Using Outlier Rule

• With the following data: ({2, 3, 4, 6, 8, 10, 14, 17, 18, 42})
  • Calculate IQR:
    $IQR = 17 - 4 = 13$
  • Calculate 1.5 × IQR:
    $1.5 \times IQR = 19.5$
  • Calculate lower bound:
    $Q1 - 1.5 \times IQR = -15.5$
  • Calculate upper bound:
    $Q3 + 1.5 \times IQR = 36.5$
  • Conclusion: 42 is an outlier.

The Modified Boxplot

• Definition: A boxplot is a graph which summarizes data using the 1.5 × IQR Rule for spotting outliers.
  • Features:
    • A central box spans from Q1 to Q3.
    • A line inside the box indicates the median (M).
    • Whiskers extending from the box to the smallest and largest observations that are not flagged as outliers.
    • Outliers are represented by symbols (e.g., “+”).

Measuring Spread: Variance

• Definition: The variance (s^2) of a set of (n) observations is calculated as:
  $s^2 = \frac{1}{n-1} \sum<em>{i=1}^{n} (x</em>i - \bar{x})^2$
• Standard Deviation (s):
  • Definition: The standard deviation is the square root of the variance.
    $s = \sqrt{s^2}$

Example of Standard Deviation

• Find the standard deviation for the dataset ({2, 3, 4, 6, 8, 10, 14, 17, 18, 42}).
  • Result: (s = 11.85).

Properties of the Standard Deviation

• The degrees of freedom is defined as (n - 1).
• The standard deviation (s) measures the spread of observations around the mean.
• When (s = 0), this indicates no spread (all observations are equal).
• Note: A few outliers can significantly inflate the value of (s), indicating it is not a robust statistic.

Choosing a Summary

• The five-number summary is preferred over mean and standard deviation when:
  • The distribution is skewed.
  • The distribution contains outliers.