Descriptive Statistics
Master the fundamental metrics that summarize and describe your data
What Are Descriptive Statistics?
Descriptive statistics are numbers that summarize and describe the characteristics of your dataset. Instead of looking at thousands of individual data points, you can understand the big picture through a few key metrics.
Think of them as the "vital signs" of your data - just like a doctor checks your temperature, blood pressure, and heart rate to understand your health, data analysts use descriptive statistics to understand their data.
Measures of Central Tendency
These tell you where the "center" or "typical value" of your data is.
Mean (Average)
The sum of all values divided by the number of values.
Formula: Mean = Sum of all values / Number of values
Example: [10, 20, 30, 40, 50]
Mean = (10+20+30+40+50) / 5 = 150 / 5 = 30
✓ Best for:
Normally distributed data without extreme outliers
✗ Watch out:
Sensitive to outliers. One extreme value can skew the mean significantly.
Median (Middle Value)
The middle value when data is sorted. Half the values are above it, half below.
Step 1: Sort the data
Example: [10, 100, 20, 50, 30] → [10, 20, 30, 50, 100]
Step 2: Find middle → Median = 30
(If even count, average the two middle values)
✓ Best for:
Skewed data or data with outliers (like income, house prices)
💡 Fun fact:
Median income is more meaningful than mean income because extreme wealth doesn't skew it!
Mode (Most Frequent)
The value that appears most often in your dataset.
Example: [5, 2, 8, 2, 9, 2, 4]
Mode = 2 (appears 3 times)
Can have multiple modes (bimodal, multimodal) or no mode
✓ Best for:
Categorical data (most popular color, most common size)
Measures of Spread (Variability)
These tell you how spread out or consistent your data is.
Range
The difference between the maximum and minimum values.
Formula: Range = Max - Min
Example: [10, 25, 30, 45, 60]
Range = 60 - 10 = 50
Simple but sensitive to outliers. Only uses two data points.
Variance
Average of squared differences from the mean. Measures how far each value is from the mean.
Steps:
- Find the mean
- Subtract mean from each value
- Square each difference
- Average the squared differences
Example: [2, 4, 6] → Mean = 4
Variance = [(2-4)² + (4-4)² + (6-4)²] / 3 = [4+0+4] / 3 = 2.67
Units are squared, which can be hard to interpret. That's why we use standard deviation!
Standard Deviation (SD)
Square root of variance. Tells you the average distance from the mean.
Formula: SD = √Variance
From example above: SD = √2.67 = 1.63
📊 The 68-95-99.7 Rule (Normal Distribution)
- • 68% of data falls within 1 SD of mean
- • 95% of data falls within 2 SD of mean
- • 99.7% of data falls within 3 SD of mean
Other Important Measures
Quartiles & Percentiles
Divide data into equal parts.
- • Q1 (25th percentile): 25% of data is below this
- • Q2 (50th percentile): Same as median
- • Q3 (75th percentile): 75% of data is below this
- • IQR (Interquartile Range): Q3 - Q1 (middle 50%)
Skewness
Measures asymmetry of distribution. Positive skew = tail on right (mean > median). Negative skew = tail on left (mean < median).
Kurtosis
Measures how "peaked" or "flat" the distribution is. High kurtosis = heavy tails and sharp peak.
Practical Example
Test scores: [65, 70, 75, 80, 85, 90, 95]
Central Tendency
- Mean: 80
- Median: 80
- Mode: None (all unique)
Spread
- Range: 30 (95-65)
- Std Dev: ~10.8
- Min: 65
- Max: 95
Interpretation: The average score is 80, with most scores falling between 69.2 and 90.8 (within 1 SD). The scores are fairly evenly distributed with no extreme outliers.
💡 Key Takeaways
- ✓Mean, median, and mode describe the center of your data
- ✓Use median for skewed data, mean for normal distributions
- ✓Standard deviation tells you how spread out the data is
- ✓Always report both central tendency AND spread for complete picture
- ✓Quartiles and IQR are useful for identifying outliers
Practice Your Skills!
Now that you understand descriptive statistics, try using our Statistics Calculator to compute these metrics for real datasets!