Chapter 3: Numerical Descriptive Measures in Statistics

3.1 Measures of Central Tendency for Ungrouped Data

Definition and Importance

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. These measures help summarize and understand large data sets by identifying a representative value.

Mean

The mean (or average) is calculated by dividing the sum of all values by the number of values in the data set. It is sensitive to every value, including outliers.

• Population Mean: $\mu = \frac{\sum x}{N}$

• Sample Mean: $\bar{x} = \frac{\sum x}{n}$

Where $\sum x$ is the sum of all values, $N$ is the population size, $n$ is the sample size, $\mu$ is the population mean, and $\bar{x}$ is the sample mean.

Example: Table 3.1 shows cash donations by eight U.S. companies in 2010. The mean donation is calculated by summing all donations and dividing by 8.

Median

The median is the value of the middle term in a data set arranged in increasing order. If the number of values is odd, the median is the middle value; if even, it is the average of the two middle values.

• Step 1: Rank the data in increasing order.

• Step 2: Identify the middle value(s).

Example: Table 3.2 lists the number of homes foreclosed in seven states in 2010. The median is the fourth value when the data is ordered.

Example: Table 3.3 shows the total compensation of 12 highest-paid CEOs in 2010. The median is the average of the 6th and 7th values in the ordered list.

Mode

The mode is the value that occurs most frequently in a data set. A data set may have no mode, one mode (unimodal), two modes (bimodal), or more than two modes (multimodal). The mode can be used for both quantitative and qualitative data.

• Unimodal: One mode

• Bimodal: Two modes

• Multimodal: More than two modes

Relationships Among the Mean, Median, and Mode

The relationship among mean, median, and mode depends on the shape of the data distribution:

• Symmetric Distribution: Mean = Median = Mode (center of the distribution)

• Right-Skewed Distribution: Mean > Median > Mode (mean is pulled right by outliers)

• Left-Skewed Distribution: Mean < Median < Mode (mean is pulled left by outliers)

3.2 Measures of Dispersion for Ungrouped Data

Definition and Importance

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

Range

The range is the difference between the largest and smallest values in a data set.

Formula: $\text{Range} = \text{Largest value} - \text{Smallest value}$

Example: Table 3.4 shows the total area of four states. The range is calculated as the difference between the largest and smallest area values.

Variance and Standard Deviation

The variance measures the average squared deviation from the mean. The standard deviation is the positive square root of the variance and is the most commonly used measure of dispersion.

• Population Variance: $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

• Sample Variance: $s^2 = \frac{\sum (x - \bar{x})^2}{n-1}$

• Population Standard Deviation: $\sigma = \sqrt{\sigma^2}$

• Sample Standard Deviation: $s = \sqrt{s^2}$

Example: Table shows baggage fee revenues for six airlines. The variance and standard deviation are calculated using the sum of values and the sum of squared values.

Example: Table shows earnings for six employees. The variance and standard deviation are calculated similarly.

3.3 Mean, Variance, and Standard Deviation for Grouped Data

Mean for Grouped Data

For grouped data, the mean is calculated using the midpoints of the classes and their frequencies.

• Population Mean: $\mu = \frac{\sum mf}{N}$

• Sample Mean: $\bar{x} = \frac{\sum mf}{n}$

Where $m$ is the class midpoint and $f$ is the class frequency.

Example: Table shows the frequency distribution of daily commuting times for 25 employees. The mean is calculated using the midpoints and frequencies.

Example: Table shows the frequency distribution of number of orders received each day. The mean is calculated similarly.

Variance and Standard Deviation for Grouped Data

The formulas for variance and standard deviation for grouped data use the midpoints and frequencies:

• Population Variance: $\sigma^2 = \frac{\sum f(m - \mu)^2}{N}$

• Sample Variance: $s^2 = \frac{\sum f(m - \bar{x})^2}{n-1}$

• Population Standard Deviation: $\sigma = \sqrt{\sigma^2}$

• Sample Standard Deviation: $s = \sqrt{s^2}$

Example: Table shows the frequency distribution of daily commuting times. The variance and standard deviation are calculated using the midpoints and frequencies.

Example: Table shows the frequency distribution of number of orders. The variance and standard deviation are calculated similarly.

Use of Standard Deviation

Chebyshev’s Theorem

Chebyshev’s theorem states that for any number $k > 1$, at least $1 - \frac{1}{k^2}$ of the data values lie within $k$ standard deviations of the mean, regardless of the data distribution.

Example: For a mean of 187 and standard deviation of 22, at least 75% of values lie between 143 and 231.

Empirical Rule

For bell-shaped (normal) distributions:

• About 68% of values lie within 1 standard deviation of the mean

• About 95% within 2 standard deviations

• About 99.7% within 3 standard deviations

Example: For a mean age of 40 and standard deviation of 12, about 95% of people are between 16 and 64 years old.

Measures of Position

Quartiles and Interquartile Range (IQR)

Quartiles divide a ranked data set into four equal parts. The first quartile (Q1) is the median of the lower half, the second quartile (Q2) is the median, and the third quartile (Q3) is the median of the upper half.

The interquartile range (IQR) is $Q3 - Q1$ and measures the spread of the middle 50% of the data.

Example: Table 3.3 (CEO compensation) is used to find quartiles and IQR.

Example: Quartile calculation for ages of employees.

Percentiles and Percentile Rank

Percentiles divide a ranked data set into 100 equal parts. The k-th percentile (Pk) is the value below which k% of the data falls.

Box-and-Whisker Plot

Definition and Construction

A box-and-whisker plot visually displays the center, spread, and skewness of a data set using the median, quartiles, and extremes (excluding outliers). The box represents the interquartile range, and the whiskers extend to the smallest and largest values within 1.5 IQR of the quartiles.