Chapter 3: Numerically Summarizing Data

3.1 Measures of Central Tendency

Measures of central tendency are statistical values that describe the center or typical value of a data set. The three main measures are the mean, median, and mode.

3.1.1 Arithmetic Mean

• Definition: The arithmetic mean (or average) is the sum of all values divided by the number of observations.

• Population Mean (μ): Uses all individuals in a population and is a parameter.

• Sample Mean (\( \overline{x} \)): Uses data from a sample and is a statistic.

• Formula for Sample Mean:

$\overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum x_i}{n}$

• Formula for Population Mean:

$\mu = \frac{\sum x_i}{N}$

• Example: For travel times 23, 36, 23, 18, 5, 26, 43, the population mean is 24.9 minutes.

3.1.2 Median

The median is the value that lies in the middle of the data when arranged in ascending order. It divides the data into two equal halves.

• Finding the Median:

  • Arrange data in ascending order.

  • If the number of observations (n) is odd, the median is the value at position \( \frac{n+1}{2} \).

  • If n is even, the median is the mean of the values at positions \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \).

• Example (Odd n): For 5, 18, 23, 23, 26, 36, 43, the median is 23.

• Example (Even n): For 5, 18, 23, 23, 26, 36, 43, 70, the median is 24.5.

3.1.3 Resistance of a Statistic

A statistic is resistant if it is not substantially affected by extreme values (outliers). The median is resistant, while the mean is not.

• Example: Adding a 130-minute commute to the previous data increases the mean more than the median.

• Distribution Shape: The relationship between mean and median can indicate skewness:

  • Mean < Median: Skewed left

  • Mean ≈ Median: Symmetric

  • Mean > Median: Skewed right

3.1.4 Mode

The mode is the most frequent observation in a data set. A set can have no mode, one mode, or more than one mode.

• Example: The mode of Vice Presidents' states of birth is New York (most frequent).

3.1.5 Summary Table: Measures of Central Tendency

3.2 Measures of Dispersion

Measures of dispersion describe the spread or variability of a data set. Common measures include range, variance, and standard deviation.

3.2.1 Range

• Definition: The range (R) is the difference between the largest and smallest values.

• Formula: $R = \text{Largest Value} - \text{Smallest Value}$

• Example: For 23, 36, 23, 18, 5, 26, 43, the range is 38 minutes.

3.2.2 Standard Deviation

• Population Standard Deviation (σ): Measures the average distance of data points from the mean.

• Formula:

$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

• Computational Formula:

$\sigma = \sqrt{\frac{\sum x_i^2 - (\sum x_i)^2 / N}{N}}$

• Sample Standard Deviation (s): Uses n-1 in the denominator (degrees of freedom).

• Formula:

$s = \sqrt{\frac{\sum (x_i - \overline{x})^2}{n-1}}$

• Computational Formula:

$s = \sqrt{\frac{\sum x_i^2 - (\sum x_i)^2 / n}{n-1}}$

• Example: For a sample (5, 26, 36), $s \approx 15.82$ minutes.

3.2.3 Variance

• Definition: The variance is the square of the standard deviation.

• Population Variance: $\sigma^2$

• Sample Variance: $s^2$

• Example: If $\sigma = 11.36$, then $\sigma^2 = 129.05$ minutes2.

3.2.4 The Empirical Rule

The Empirical Rule applies to bell-shaped (normal) distributions:

• About 68% of data within 1 standard deviation of the mean

• About 95% within 2 standard deviations

• About 99.7% within 3 standard deviations

3.2.5 Chebyshev’s Inequality

Chebyshev’s Inequality applies to any data set, regardless of shape:

• At least $1 - \frac{1}{k^2}$ of the data lies within k standard deviations of the mean, for $k > 1$.

3.3 Measures from Grouped Data

When only grouped (frequency) data is available, we can approximate the mean and standard deviation.

3.3.1 Approximating the Mean

• Formula:

$\mu = \frac{\sum x_i f_i}{\sum f_i}$ $\overline{x} = \frac{\sum x_i f_i}{\sum f_i}$

3.3.2 Weighted Mean

• Definition: The weighted mean accounts for values with different levels of importance (weights).

• Formula:

$\overline{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

3.3.3 Approximating the Standard Deviation

• Formula:

$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2 f_i}{\sum f_i}}$ $\sigma = \sqrt{\frac{\sum x_i^2 f_i - (\sum x_i f_i)^2 / \sum f_i}{\sum f_i}}$

3.4 Measures of Position and Outliers

Measures of position describe the relative standing of a value within a data set. Outliers are values that are unusually far from the rest of the data.

3.4.1 z-Scores

• Definition: The z-score measures how many standard deviations a value is from the mean.

• Formulas:

$z = \frac{x - \mu}{\sigma}$ (population) $z = \frac{x - \overline{x}}{s}$ (sample)

• Example: Comparing heights using z-scores.

3.4.2 Percentiles

• Definition: The kth percentile is the value below which k% of the data falls.

3.4.3 Quartiles

• Definition: Quartiles divide data into four equal parts.

• Q1: 25th percentile, Q2: 50th percentile (median), Q3: 75th percentile

3.4.4 Interquartile Range (IQR)

• Definition: The IQR is the range of the middle 50% of the data: $IQR = Q_3 - Q_1$

3.4.5 Identifying Outliers

• Fences: Outliers are values outside the lower and upper fences:

  • Lower Fence = $Q_1 - 1.5 \times IQR$

  • Upper Fence = $Q_3 + 1.5 \times IQR$

• Values outside these fences are considered outliers.

3.5 The Five-Number Summary and Boxplots

The five-number summary provides a concise description of a data set, and boxplots visually display this summary.

3.5.1 Five-Number Summary

• Consists of: Minimum, Q1, Median (Q2), Q3, Maximum

• Example: For credit card interest rates: 6.5%, 12.0%, 13.6%, 14.4%, 14.5%

3.5.2 Boxplots

• Boxplots display the five-number summary and highlight outliers.

• Steps:

  • Draw a box from Q1 to Q3, with a line at the median.

  • Draw whiskers to the smallest and largest values within the fences.

  • Mark outliers with asterisks.

• Boxplots can be used to compare distributions and identify skewness.