Learning Objective
Understand the concepts of descriptive statistics, including measures of central tendency, variability, and the characteristics of distributions, and interpret these measures in context.
Distributions
Statistics often view the world as distributions of data. These distributions are summarized by:
- Central tendency – the “center” of the data
- Variability – how spread out the data are
The most important distribution is the normal (Gaussian) curve, a symmetric “bell-shaped” curve:
- The left and right sides are mirror images.
- Many real-world measurements (e.g., heights, test scores) approximate this curve.
Measures of Central Tendency
Central tendency identifies a single value that represents the middle of a dataset. Common measures include:
Mean:
Example: Dataset: 3, 6, 7, 7, 9, 10, 12, 15, 16
Median (Md):
Middle value when data are ordered; 50th percentile
Mode:
Most frequently occurring value
Activity
Skewed Distributions:
- Positive skew: Tail to the right; mean > median
- Negative skew: Tail to the left; median > mean
In skewed distributions, the median is often a better measure of central tendency than the mean.
Measures of Variability
Variability measures how spread out the data are.
Range:
Simple but unstable; sensitive to extreme values.
Standard Deviation (SD)
Compute deviations from the mean:
Square deviations:
Average squared deviations:
Take the square root:
Variance (s2s^2)
It is the square of the SD:
Activity
Normal Curve and SD:
- Within 1 SD: 68% of cases
- Within 2 SDs: 95.5% of cases
- Within 3 SDs: 99.7% of cases
On exams, you are not expected to calculate SD or variance, but you should understand their relationship to the normal curve.
Summary
- Central tendency = mean, median, mode
- Variability = range, SD, variance
- Normal distribution = symmetric, with predictable proportions of data within SDs
- Skewed distributions require a careful choice of central tendency
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