Mean (Average)

• Calculation: Sum the values, then divide by the number of values.
• Formula: Mean = (Sum of all values) / (Number of values)
• Usage: The mean works well for symmetric data without extreme outliers.

Median

• Calculation: Sort the data and take the middle value. For an even count, average the two middle values.
• Usage: Prefer the median for skewed data or data with outliers because extreme values do not pull it toward a tail.

Mode

• Calculation: The mode is the most frequent value.
• Usage: Use it for categorical data or whenever the most common value matters.

Comparing Means of Distributions

• Never compare means alone. Check each distribution's shape, spread, and outliers.

Impact of Outliers

• Removing an outlier: A sufficiently extreme value can move the mean substantially when removed.
• Increasing an outlier: Moving an outlier farther from the data pulls the mean with it.

Mean as the Balance Point

• The mean acts as the distribution's balance point. In a symmetric distribution, values balance evenly around it.

Choose mean, median, or mode from the distribution in front of you, not from habit.

Interquartile Range (IQR)

• Calculation: Subtract the 25th percentile (Q1) from the 75th percentile (Q3).
• Formula: IQR = Q3 - Q1
• Use case: IQR measures the middle 50% and supports common outlier rules.

Range

• Calculation: Subtract the minimum from the maximum.
• Formula: Range = Maximum value - Minimum value
• Use case: The range is quick but extremely sensitive to outliers.

Variance

• Population variance: The average squared distance from the mean.
• Formula for Population Variance ((\sigma^2)): (\sigma^2 = \frac{\sum (X - \mu)^2}{N}) where (X) is each value, (\mu) is the mean, and (N) is the number of values.
• Sample variance: Divide by (N-1), not (N), to estimate population variance from a sample.
• Use case: Variance quantifies squared distance from the mean.

Standard Deviation

• Calculation: Take the square root of the variance.
• Formula for Population Standard Deviation ((\sigma)): (\sigma = \sqrt{\sigma^2})
• Formula for Sample Standard Deviation (s): (s = \sqrt{\frac{\sum (X - \bar{X})^2}{N-1}})
• Use case: Standard deviation measures spread around the mean in the data's original units.

Calculating Standard Deviation Step by Step

1. Find the Mean: Sum all the data points and divide by the number of points.
2. Calculate Each Point's Deviation from the Mean: Subtract the mean from each data point.
3. Square Each Deviation: Square each result from step 2.
4. Sum the Squared Deviations: Add up all the squared deviations.
5. Divide by (N-1) for a Sample, or (N) for a Population: This gives the variance.
6. Take the Square Root of the Variance: This gives the standard deviation.

Understanding the Concepts

• Population variance: Measures squared spread around the population mean.
• Population standard deviation: Measures that spread in the data's original units.
• Interpretation: Variance uses squared units; standard deviation converts the result back to the original units.

Pick the measure that matches the analysis and the units you need to interpret.

Dividing by ( n - 1 ) in Variance Calculation

Sample variance uses ( n - 1 ), rather than ( n ), in the denominator. This is Bessel's correction.

• Reason for ( n - 1 ): Estimating variance around the sample mean with ( n ) systematically underestimates population variance. Dividing by ( n - 1 ) removes that bias.
• Concept: Estimating the mean from the same sample imposes one constraint on the deviations: they must sum to zero.
• Degrees of freedom: Once the first ( n - 1 ) deviations are fixed, the final deviation is determined. Only ( n - 1 ) values remain free to vary.

Box and Whisker Plots

Box-and-whisker plots visualize a distribution's five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.

• Components of a Box Plot:

  1. Box: The box spans from Q1 to Q3, representing the interquartile range (IQR).
  2. Median: A line across the box indicates the median of the dataset.
  3. Whiskers: Lines extending from the box (whiskers) typically extend to the minimum and maximum values within 1.5 * IQR from the Q1 and Q3. Values beyond this are plotted as outliers.
  4. Outliers: Points beyond the whiskers are considered outliers and are plotted individually.

• Interpreting a Box Plot:

  • Box length shows the spread of the middle 50%.
  • The median's position within the box can suggest asymmetry, but it does not establish skew by itself.
  • Points beyond the whiskers flag potential outliers for inspection.

• Use case: Box plots compare center and spread across groups without drawing every observation.

Use box plots for a compact comparison, then inspect the underlying distribution before drawing conclusions.