Descriptive Statistics: Measures of Central Tendency, Variation, and Relative Position

Descriptive Statistics

Introduction

Descriptive statistics are essential tools in business statistics, providing methods to summarize, organize, and interpret data. The main measures include central tendency, variability, and relative position, each offering unique insights into data sets.

Measures of Central Tendency

The Mean (Arithmetic Mean)

The mean is the most common measure of central tendency, calculated by summing all values and dividing by the number of observations. It represents the average value in a data set.

• Formula (Sample Mean): $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

• Formula (Population Mean): $\mu = \frac{\sum_{i=1}^{N} x_i}{N}$

• Example: For data set {87.2, 118.9, 76.2, 107.7, 61.5}, the mean is $\bar{x} = \frac{87.2 + 118.9 + 76.2 + 107.7 + 61.5}{5} = 90.3$

Weighted Mean

The weighted mean assigns different weights to values, useful when some data points contribute more significantly than others.

• Formula: $\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$

• Example: If exam, project, and homework scores are weighted 50%, 35%, and 15% respectively, the weighted mean is calculated as $\frac{(94 \times 0.5) + (92 \times 0.35) + (100 \times 0.15)}{0.5 + 0.35 + 0.15} = 94.2$

The Median

The median is the middle value when data are ordered. If the number of observations is even, it is the average of the two middle values.

• Formula (Index Point): $i = 0.5n$

• Example: For sorted data {26, 28, 31, 39, 43, 45, 45, 50, 57, 62}, the median is the average of the 5th and 6th values: $(43 + 45)/2 = 44$

The Mode

The mode is the value that appears most frequently in a data set. Data can be unimodal, bimodal, or have no mode.

• Example: In {6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 14, 14}, the mode is 8 (appears 5 times).

Choosing the Appropriate Measure

• Mean: Easy to calculate, but sensitive to outliers.

• Median: Not affected by outliers, better for skewed data.

• Mode: Useful for categorical data, may not always exist or may be multiple.

Measures of Variation

Range

The range is the difference between the highest and lowest values in a data set.

• Formula: $R = \text{Highest value} - \text{Lowest value}$

• Limitation: Highly affected by outliers and does not consider data distribution shape.

Variance and Standard Deviation

Variance measures the average squared deviation from the mean, while standard deviation is its square root, representing spread in the same units as the data.

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

• Population Variance: $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$

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

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

Coefficient of Variation (CV)

The coefficient of variation expresses the standard deviation as a percentage of the mean, allowing comparison of variability between data sets with different units or means.

• Sample CV: $CV = \frac{s}{\bar{x}} \times 100$

• Population CV: $CV = \frac{\sigma}{\mu} \times 100$

Using the Mean and Standard Deviation Together

Shapes of Frequency Distribution

Data distributions can be symmetric, left-skewed, or right-skewed. Skewness measures asymmetry, while kurtosis measures the peakedness of the distribution.

• Symmetric: Mean = Median

• Left-skewed: Mean < Median

• Right-skewed: Mean > Median

Quality Control Example: Histograms

Histograms visually display the distribution of data and help identify the mean, standard deviation, and conformity to specifications.

The z-Score

Definition and Calculation

The z-score indicates how many standard deviations a value is from the mean, standardizing different data sets for comparison.

• Population: $z = \frac{x - \mu}{\sigma}$

• Sample: $z = \frac{x - \bar{x}}{s}$

• Interpretation: A z-score < -3 or > +3 is considered an extreme outlier.

The Empirical Rule

For bell-shaped (normal) distributions:

• Approximately 68% of values fall within ±1 standard deviation from the mean

• Approximately 95% within ±2 standard deviations

• Approximately 99.7% within ±3 standard deviations

Chebyshev’s Theorem

For any distribution (not just normal), at least $100 \times (1 - 1/z^2)$% of values fall within z standard deviations from the mean, for z > 1.

• At least 75% within ±2 standard deviations

• At least 89% within ±3 standard deviations

• At least 94% within ±4 standard deviations

Measures of Relative Position

Percentiles

Percentiles indicate the percentage of data values below a certain point. The pth percentile is the value below which p% of the data fall.

• Index Point Formula: $i = \frac{p}{100} n$

• If i is not a whole number, round up; if i is whole, average the ith and (i+1)th values.

Quartiles

Quartiles divide data into four equal parts:

• Q1: 25th percentile

• Q2: 50th percentile (median)

• Q3: 75th percentile

Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of data and is not influenced by outliers.

• Formula: $IQR = Q3 - Q1$

Box-and-Whisker Plots

A boxplot visually displays the five-number summary (minimum, Q1, median, Q3, maximum) and identifies outliers.

• Upper Limit: $Q3 + 1.5 \times IQR$

• Lower Limit: $Q1 - 1.5 \times IQR$

• Values outside these limits are considered outliers.

Excel/PHStat Applications

Descriptive Statistics in Excel

Excel provides tools for calculating mean, median, mode, standard deviation, variance, percentiles, quartiles, and creating boxplots.

• Use Data Analysis → Descriptive Statistics for summary statistics.

• Functions: =AVERAGE(), =MEDIAN(), =MODE.SNGL(), =STDEV.S(), =VAR.S(), =PERCENTILE.EXC(), =QUARTILE.EXC()

Summary

• Measures of Central Tendency: Mean, Median, Mode

• Measures of Variation: Range, Variance, Standard Deviation, Coefficient of Variation

• Measures of Location: Percentile, Quartile, Box & Whisker Plot