Lesson 90: Descriptive Statistics
Descriptive statistics summarize and describe data sets. They help us understand the central tendency (typical value), spread (variability), and shape of data. Key measures include mean, median, mode, range, variance, and standard deviation.
Measures of Central Tendency
- Mean (average): Sum of data values ÷ number of values.
- Median: Middle value when data is ordered (or average of two middle values).
- Mode: Value that appears most often in the data set.
Measures of Spread
- Range: Maximum − Minimum.
- Variance: Average squared deviation from the mean.
- Standard Deviation: Square root of variance. Interpreted as the “typical†distance from the mean.
Formulas
For data values \(x_1, x_2, ..., x_n\) with mean \(\bar{x}\):
\( \text{Variance } (s^2) = \frac{\sum (x_i - \bar{x})^2}{n} \)
\( \text{Standard Deviation } (s) = \sqrt{s^2} \)
Examples
Example 1 — Small Data Set:
Data: 2, 4, 6, 8, 10 Mean = (2+4+6+8+10)/5 = 6 Median = 6 Mode = none (all unique) Range = 10 − 2 = 8 Variance = [(2−6)²+(4−6)²+(6−6)²+(8−6)²+(10−6)²]/5 = 8 Standard Deviation = √8 ≈ 2.83
Example 2 — Mode:
Data: 3, 5, 5, 7, 9 Mode = 5, since it appears twice.
Practice Problems
- Find the mean, median, and mode of: 12, 15, 18, 12, 20.
- Compute the range, variance, and standard deviation for: 5, 10, 15, 20.
- Which measure of central tendency is best when data has extreme outliers? (Mean, median, or mode?)
- Find the variance and standard deviation for: 2, 2, 4, 4.
- Explain why variance and standard deviation are more informative than the range.