Understanding Statistical Measures
Mean (Average)
The arithmetic mean is calculated by summing all numbers and dividing by the count:
Example: [10, 20, 30] โ (10+20+30) รท 3 = 20
When to use: Best for normally distributed data without extreme outliers. Commonly used for grades, measurements, and general statistics.
Median
The median is the middle value when numbers are sorted. For even counts, it's the average of the two middle values.
Even count: [10, 20, 30, 40] โ median = (20+30) รท 2 = 25
When to use: Better than mean for skewed data or when outliers exist. Median income is more representative than mean income because billionaires skew the average.
Mode
The mode is the most frequently occurring value(s). A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal).
[1, 2, 2, 3, 3, 4] โ modes = 2 and 3 (bimodal)
[1, 2, 3, 4, 5] โ no mode (all appear once)
When to use: Best for categorical data or finding the most common value. Example: most popular shoe size, most common test score.
Standard Deviation (ฯ)
Standard deviation measures how spread out numbers are from the mean. Low SD = consistent data, high SD = variable data.
High SD Example: Test scores [50, 75, 100] โ SD โ 25 (high variability)
68-95-99.7 Rule: In a normal distribution, ~68% of data falls within 1 SD of mean, ~95% within 2 SD, ~99.7% within 3 SD.
Variance (ฯยฒ)
Variance is the average of squared differences from the mean. Standard deviation is simply the square root of variance.
Variance = [(2-4)ยฒ + (4-4)ยฒ + (6-4)ยฒ] รท 3 = [4+0+4] รท 3 = 2.67
Standard Deviation = โ2.67 = 1.63
Real-World Applications
- Academic: Calculate GPA, test score averages, grade distributions
- Business: Average sales, revenue trends, customer ratings
- Sports: Batting averages, scoring averages, performance metrics
- Finance: Investment returns, stock price movements, portfolio analysis
- Research: Scientific data analysis, survey results, experimental measurements
- Health: BMI calculations, blood pressure tracking, medication effectiveness
๐ก Pro Tip
When analyzing data, always look at mean, median, and standard deviation together. If mean and median differ significantly, your data may be skewed. High standard deviation indicates you should investigate why values vary so much.