Statistics Infographic (Simplified)
Central Tendency, Spread, Boxplots, & More
Example Dataset (8 values):
Measures of Central Tendency
Central tendency measures show where data values cluster (the “center” of the dataset).
Mean (Average)
Add all values, then divide by how many there are.
Example:
(4 + 7 + 10 + 12 + 12 + 15 + 18 + 22) ÷ 8 = 12.5
Pros:
Uses all data points. Good for further math (e.g., variance).
Cons:
Easily skewed by extreme values (outliers).
Median
The middle value when all data points are sorted.
Example:
Sorted data: 4, 7, 10, 12, 12, 15, 18, 22
Median = (12 + 12) / 2 = 12
Pros:
Not pulled up or down by extreme values.
Cons:
Ignores distance between values.
Mode
The most frequently occurring value(s).
Example:
4, 7, 10, 12, 12, 15, 18, 22
12 appears most often.
Pros:
Helpful for categorical data or to find the most common value.
Cons:
Some datasets have no mode or multiple modes.
Measures of Spread (Dispersion)
These show how “spread out” or varied the data is.
Range
The distance between the smallest and largest value.
Example:
22 - 4 = 18
Pros:
Simple and fast.
Cons:
Only looks at two values, ignores the rest.
Variance
Measures how far each value is from the mean, on average (in squared units).
Example (simplified):
Mean = 12.5. Sum of squared differences = 267.
Variance = 267 / 8 = 33.375 (population).
Pros:
Uses all data; basis for standard deviation.
Cons:
Units are squared; less intuitive to read.
Standard Deviation
The square root of the variance, showing average distance from the mean in original units.
Example:
√33.375 ≈ 5.78 (population).
Pros:
Easier to interpret than variance; same units as data.
Cons:
Still affected by outliers.
Interquartile Range (IQR)
The range of the middle 50% of data (between Q1 and Q3).
Example:
Q1 ≈ 7.75, Q3 ≈ 16.5 → IQR ≈ 8.75
Pros:
Resistant to outliers; focuses on central spread.
Cons:
Ignores the top and bottom 25% of the data.
Distribution Shapes
Knowing the shape helps you pick suitable measures and tests.
Normal (Symmetric)
Mean ≈ Median ≈ Mode
Use Mean & Standard Deviation
Right-Skewed
Choose Median & IQR for summary.
Left-Skewed
Also use Median & IQR.
AVERAGE, MEDIAN, MODE.SNGL, SKEW, QUARTILE.INC, CONFIDENCE, etc. all help analyze data shape and spread.
Extended Dataset (12 values):
5-Number Summary & Quartiles
The 5-number summary is a quick snapshot of a dataset's spread, focusing on key points: Minimum, Q1, Median, Q3, and Maximum.
Quartiles
Quartiles split the sorted data into four equal parts:
- Q1 (25th percentile): Lower quartile
- Q2 (50th percentile, Median): Middle
- Q3 (75th percentile): Upper quartile
Example (Sorted Data):
4, 7, 10, 12, 12, 15, 18, 22, 25, 28, 30, 42
Q1 = ~10, Median(Q2) = 15, Q3 = ~26.5
Note: Different tools may calculate quartiles with slightly different methods.
5-Number Summary
A set of five key values that help you see the spread at a glance:
Calculation Example:
Minimum = 4
Q1 ≈ 10
Median = 15
Q3 ≈ 26.5
Maximum = 42
So the 5-number summary is {4, 10, 15, 26.5, 42}.
Interpretation:
Minimum & Maximum show the extreme ends. Q1 & Q3 mark the middle 50%. Median splits data in half.
Pros:
Offers a concise overview; good for comparing multiple sets.
Cons:
Doesn't reveal all data details (like multiple peaks).
Boxplots (Box-and-Whisker Plots)
A visual tool to display the 5-number summary and spot possible outliers.
Boxplot Components
The box shows the IQR, the line inside is the median, and whiskers extend to typical extremes.
(Scale here is just an illustration.)
1. Box = IQR
The box stretches from Q1 to Q3.
2. Median Line
The vertical line inside the box is the median.
3. Whiskers
Extend to data points within 1.5×IQR of the box edges.
4. Outliers
Points beyond the whiskers are potential outliers.
Use “Box and Whisker” from Insert > Charts (newer Excel). Older versions require manual steps.
Identifying Outliers
Outliers often fall outside:
Example:
Q1 = 10, Q3 = 26.5, IQR = 16.5
Lower boundary = 10 - 24.75 = -14.75
Upper boundary = 26.5 + 24.75 = 51.25
No data point beyond these, so no outliers here.
Interpreting Boxplot Shapes
Boxplots let you see skew, spread, and possible extreme values quickly.
| Boxplot Shape | Characteristics | Interpretation |
|---|---|---|
| Symmetric | Median is centered, whiskers about equal | Distribution is likely symmetrical (maybe normal) |
| Right-Skewed | Longer whisker to the right | Mean > Median; positive skew |
| Left-Skewed | Longer whisker to the left | Mean < Median; negative skew |
| Short Box | Small IQR | Middle 50% of data is tightly packed |
| Wide Box | Large IQR | Middle 50% is more spread out |
| Many Outliers | Multiple points outside whiskers | Could show heavy tails or real anomalies |
Sample vs. Population
If you only have a sample from a larger population, formulas adjust to avoid bias.
| Measure | Sample Formula | Excel Function |
|---|---|---|
| Variance | s² = ∑(x - x̄)² / (n - 1) | VAR.S(range) |
| Std. Deviation | s = √[∑(x - x̄)² / (n - 1)] | STDEV.S(range) |
| Quartiles | Interpolation often used; (n+1) in position | QUARTILE.INC(range, quart) |
| Percentiles | Interpolates ranks | PERCENTILE.INC(range, k) |