Numerical Methods for Describing Data Distributions

Selecting Numerical Summaries

• Measures of center describe typical values along a number line.
• Measures of variability describe how much individual values differ from one another.
• Distribution shape dictates the choice of numerical summaries:

• Symmetric distributions: Described using sample mean ($\bar{x}$) and sample standard deviation ($s$).
• Skewed distributions or distributions with outliers: Described using sample median and interquartile range ($iqr$).

Describing Symmetric Distributions

• Sample Mean ($\bar{x}$): The arithmetic average of all observations in a sample. $\bar{x} = \frac{\sum x}{n}$
• Population Mean ($\mu$): The arithmetic average of all values in an entire population.
• Deviations from the Mean: Difference between an observation and the sample mean. $(x - \bar{x})$
• Sample Variance ($s^2$): Sum of squared deviations from the mean divided by $n - 1$. $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$
• Sample Standard Deviation ($s$): Square root of the sample variance. $s = \sqrt{s^2} = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$
• Population Variance ($\sigma^2$) and Population Standard Deviation ($\sigma$): Parameters representing variability for an entire population.

Describing Skewed Distributions and Handling Outliers

• Sample Median: The middle value when data are listed in order from smallest to largest.
  • If sample size $n$ is odd: The single middle value.
  • If sample size $n$ is even: The average of the two middle values.
• Quartiles:
  • Lower Quartile: Median of the lower half of the data set.
  • Upper Quartile: Median of the upper half of the data set.
  • If $n$ is odd, the overall median is excluded from both halves when calculating quartiles.
• Interquartile Range ($iqr$): Measuring variability for middle 50% of data, resistant to outliers. $iqr = \text{upper quartile} - \text{lower quartile}$

Boxplots and Five-Number Summaries

• Five-Number Summary: Minimum value, lower quartile, median, upper quartile, and maximum value.
• Outlier Rule (1.5 IQR Rule): An observation is officially an outlier if it is:
  • Greater than $\text{upper quartile} + 1.5(iqr)$
  • Less than $\text{lower quartile} - 1.5(iqr)$
• Modified Boxplots: Display individual outliers as plotted dots and extend whiskers to the nearest non-outlier values.
• Comparative Boxplots: Display multiple boxplots on a shared numerical scale to compare center, variability, and shape across groups.

Measures of Relative Standing: z-Scores and Percentiles

• z-Score: Represents the number of standard deviations a data value $x$ lies from the mean $\bar{x}$. $z\text{-score} = \frac{\text{data value} - \text{mean}}{\text{standard deviation}} = \frac{x - \bar{x}}{s}$
• Standardizing: The process of subtracting the mean and dividing by the standard deviation.
• Empirical Rule: Applies to mound-shaped and approximately symmetric distributions:
  • Approximately $68\%$ of observations fall within $1$ standard deviation ($\bar{x} \pm 1s$).
  • Approximately $95\%$ of observations fall within $2$ standard deviations ($\bar{x} \pm 2s$).
  • Approximately $99.7\%$ of observations fall within $3$ standard deviations ($\bar{x} \pm 3s$).

• Percentiles: The $r$\text{th percentile} is the value such that $r\%$ of the data fall at or below it.
  • Lower quartile = $25$\text{th percentile}
  • Median = $50$\text{th percentile}
  • Upper quartile = $75$\text{th percentile}

Key Pitfalls and Common Mistakes

• Do not summarize numerically coded categorical variables (such as zip codes or class standings) using mean, median, standard deviation, or IQR.
• Never rely solely on measures of center; always combine them with measures of variability and distribution shape.
• Avoid using mean and standard deviation on heavily skewed distributions or data sets containing notable outliers.
• Do not confuse the variability of values along the x-axis with the variation in frequency bar heights.
• Do not apply the Empirical Rule to distributions that are skewed or non-mound-shaped.