Statistics Chapter 3: Measures of Center, Variation, and Relative Standing

Chapter 3: Descriptive Statistics

3.1 Measures of Center

Measures of center are statistical values that describe the central tendency of a dataset. They help summarize a large set of data with a single representative value.

• Mean: The arithmetic average of a dataset, calculated by summing all data values and dividing by the number of values. Formula: Sample Mean: $\bar{x} = \frac{\sum x_i}{n}$ Population Mean: $\mu = \frac{\sum x_i}{N}$ Example: For data values 2, 5, 9, 7, the mean is $(2+5+9+7)/4 = 5.75$.

• Median: The middle value when data are ordered. If the number of data points is even, the median is the average of the two middle values.

• Mode: The value(s) that occur most frequently in the dataset. There can be no mode, one mode (unimodal), or multiple modes (multimodal).

• Midrange: The value halfway between the maximum and minimum values. Formula: $\text{Midrange} = \frac{\text{Max} + \text{Min}}{2}$

Resistant Statistics: Measures that are not significantly affected by extreme values (outliers). The median and mode are resistant; the mean and midrange are not.

Pros & Cons of Measures of Center

Statistic	Pros	Cons
Mean	Uses all data values	Affected by outliers
Median	Resistant to outliers	Ignores magnitude of values
Mode	Easy to identify	May not exist or may be multiple
Midrange	Simple calculation	Highly affected by outliers

Weighted Mean

Used when data values have different weights (e.g., GPA calculation).

Formula: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$ Example: Calculating GPA using course grades and credit hours.

Mean of a Frequency Distribution

When data are grouped into classes, the mean can be approximated using class midpoints and frequencies.

Formula: $\bar{x} = \frac{\sum f x}{\sum f}$ Example: For time intervals and frequencies, multiply each midpoint by its frequency, sum, and divide by total frequency.

Median of a Frequency Distribution

For grouped data, the median is estimated using:

Formula: $\text{Median} = \text{(lower limit of median class)} + \left( \frac{\frac{n+1}{2} - \text{sum of frequencies before median class}}{\text{frequency of median class}} \right) \times \text{class width}$ Example: Find the class containing the median and apply the formula.

Computing Measures of Center using Excel

Statistic	Formula	Example
Mean	=AVERAGE(CELL RANGE)	=AVERAGE(A1:A18)
Median	=MEDIAN(CELL RANGE)	=MEDIAN(A1:A18)
Mode	=MODE.MULT(CELL RANGE)	=MODE.MULT(A1:A18)
Midrange	=(MAX(CELL RANGE)+MIN(CELL RANGE))/2	=(MAX(A1:A18)+MIN(A1:A18))/2

3.2 Measures of Variation

Measures of variation describe the spread or dispersion of data values. They help understand how much the data values differ from each other.

• Range: The difference between the maximum and minimum values. Formula: $\text{Range} = \text{Max} - \text{Min}$

• Standard Deviation: Measures the average distance of data values from the mean. Larger values indicate greater variability. Sample Standard Deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ Population Standard Deviation: $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$

• Variance: The square of the standard deviation. Sample Variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ Population Variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

Estimating Standard Deviation: Range Rule of Thumb

Standard deviation can be roughly estimated as:

Formula: $s \approx \frac{\text{Range}}{4}$ Example: For test scores with a range of 40, estimated $s \approx 10$.

Significantly Low and High Values

Values more than two standard deviations from the mean are considered significantly low or high.

Category	Range
Significantly Low	$< \mu - 2\sigma$
Not Significant	$\mu - 2\sigma$ to $\mu + 2\sigma$
Significantly High	$> \mu + 2\sigma$

The Empirical Rule

For 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

Calculating Measures of Variation in Excel

Measure	Formula	Example
Range	=MAX(CELL RANGE)-MIN(CELL RANGE)	=MAX(A1:A18)-MIN(A1:A18)
Sample Standard Deviation	=STDEV.S(CELL RANGE)	=STDEV.S(A1:A18)
Sample Variance	=VAR.S(CELL RANGE)	=VAR.S(A1:A18)
Population Standard Deviation	=STDEV.P(CELL RANGE)	=STDEV.P(A1:A18)
Population Variance	=VAR.P(CELL RANGE)	=VAR.P(A1:A18)

3.3 Measures of Relative Standing & Boxplots

Measures of relative standing indicate the position of a data value relative to other values in the dataset. Common measures include z-scores, percentiles, and quartiles.

• Z-Score: Indicates how many standard deviations a value is from the mean. Formula: $z = \frac{x - \mu}{\sigma}$ Example: For a baby weighing 4000g, with mean 3152g and $\sigma = 693.4g$, $z = \frac{4000 - 3152}{693.4} \approx 1.22$.

• Percentiles: Divide data into 100 equal parts. The pth percentile is the value below which p% of the data fall. Formula for Percentile of Value: $\text{Percentile of Value} = \frac{\text{number of values below } x}{n} \times 100$

• Quartiles: Divide data into four equal parts.

  • Q1: 25th percentile

  • Q2: 50th percentile (median)

  • Q3: 75th percentile

Interquartile Range (IQR)

Measures the spread of the middle 50% of data.

Formula: $\text{IQR} = Q_3 - Q_1$ Example: For test scores, if $Q_1 = 60$ and $Q_3 = 90$, then $IQR = 30$.

Five-Number Summary

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

Boxplots

Graphical representation of the five-number summary. Useful for visualizing spread, center, and outliers.

Checking for Outliers Using Quartiles

• Lower fence: $Q_1 - 1.5 \times IQR$

• Upper fence: $Q_3 + 1.5 \times IQR$

• Values outside these fences are considered outliers.

Descriptive Statistics in Excel

Excel functions can be used to compute mean, median, mode, range, standard deviation, variance, quartiles, and to construct boxplots.

Summary Table: Key Formulas

Measure	Formula (LaTeX)
Sample Mean	$\bar{x} = \frac{\sum x_i}{n}$
Population Mean	$\mu = \frac{\sum x_i}{N}$
Sample Standard Deviation	$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$
Population Standard Deviation	$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$
Sample Variance	$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$
Population Variance	$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$
Midrange	$\frac{\text{Max} + \text{Min}}{2}$
Range	$\text{Max} - \text{Min}$
Z-Score	$z = \frac{x - \mu}{\sigma}$
Interquartile Range	$Q_3 - Q_1$

Additional info: Some context and examples were expanded for clarity and completeness.