Statistics Calculator
5 Number Summary Calculator
Last Updated On
Summarize any dataset instantly. The five-number summary provides a robust look at the spread and center of your data, helping you find quartiles, the median, and potential outliers.
5 Number Summary
Analyze distribution and spread of data
Results
Box Plot
Sorted Dataset:
5, 7, 8, 12, 15, 18, 22, 29, 31, 34
Reading the Five-Number Summary as a Box Plot
The box spans the middle 50% of the data, from Q1 to Q3, with the median inside it. The whiskers reach the minimum and maximum. Values beyond the 1.5 × IQR fences are flagged as outliers.
Understanding the 5 Number Summary
In descriptive statistics, a five-number summary simplifies a large block of numbers into five key markers: the minimum, Q1 (25th percentile), median (50th percentile), Q3 (75th percentile), and maximum.
By looking at these values, you can instantly see the range, identify skewness, and prepare data for constructing box-and-whisker plots.
How Outliers Are Detected
The Interquartile Range (IQR) represents the width of the middle 50% of your data.
This calculator flags outliers automatically using the standard 1.5 × IQR rule. If a data point is below Q1 - (1.5 × IQR) or above Q3 + (1.5 × IQR), it is heavily skewed from the majority of the dataset.
Exclusive vs. Inclusive Quartile Methods
Comparing the calculated quartiles for the dataset [1, 3, 5, 7, 9, 11, 13] using both standard styles:
| Statistical Parameter | Exclusive Method (Used Here) | Inclusive Method |
|---|---|---|
| Median (Q2) | 7.00 | 7.00 |
| First Quartile (Q1) | 3.00 (from [1, 3, 5]) | 4.00 (from [1, 3, 5, 7]) |
| Third Quartile (Q3) | 11.00 (from [9, 11, 13]) | 10.00 (from [7, 9, 11, 13]) |
| Interquartile Range (IQR) | 8.00 | 6.00 |
The 5 Statistics Definitions
The five key descriptive statistical markers explained:
| Marker | Formula / Determination | Percentile Represented |
|---|---|---|
| Minimum | Lowest numeric value in sorted data | 0th percentile |
| First Quartile (Q1) | Median of the lower half of dataset | 25th percentile |
| Median (Q2) | Middle value (average of middle 2 if even count) | 50th percentile |
| Third Quartile (Q3) | Median of the upper half of dataset | 75th percentile |
| Maximum | Highest numeric value in sorted data | 100th percentile |
Example Calculations
Example Scenario 1 — Small Dataset, No Outliers
Data: 1, 2, 5, 6, 7, 9, 12, 15
Sorted: 1, 2, 5, 6, 7, 9, 12, 15
Min = 1, Max = 15
Median (middle two 6, 7) = 6.5
Lower half: 1, 2, 5, 6 → Q1 = 3.5
Upper half: 7, 9, 12, 15 → Q3 = 10.5
Example Scenario 2 — Dataset With Outliers
Data: 10, 12, 15, 14, 13, 90
Sorted: 10, 12, 13, 14, 15, 90
Q1 = 12, Q3 = 15
IQR = 15 - 12 = 3
Upper limit = Q3 + (1.5 × 3) = 19.5
Since 90 > 19.5, 90 is flagged as an outlier.
Frequently Asked Questions
- What is a 5 number summary?
- It is a set of descriptive statistics that gives information about your dataset's center and spread. The five numbers are: Minimum, First Quartile (Q1), Median, Third Quartile (Q3), and Maximum.
- How are quartiles calculated?
- This calculator uses the exclusive method (common in TI-83 calculators and general statistics). First, find the median of the dataset. Then, find the median of the lower half (Q1) and upper half (Q3), excluding the main median from both halves if the count is odd.
- What is the IQR?
- The Interquartile Range (IQR) answers how spread out the middle 50% of your data is. It is calculated by subtracting Q1 from Q3 (IQR = Q3 - Q1).
- How are outliers found?
- Outliers are values that fall significantly outside the expected range. We use the 1.5 × IQR rule. Any value lower than (Q1 - 1.5 × IQR) or higher than (Q3 + 1.5 × IQR) is considered an outlier.
- Why is it called a five-number summary?
- Because it uses exactly five representative values—Minimum, Q1, Median, Q3, and Maximum—to capture the full distribution shape and scale of any dataset.
- How does the inclusive method differ from the exclusive method for quartiles?
- The exclusive method excludes the median when dividing the dataset into lower and upper halves to calculate Q1 and Q3, whereas the inclusive method includes the median in both halves. Exclusive is standard for TI-83 calculators and MM text.
- What is a box and whisker plot?
- A box and whisker plot is a graphical rendering of a five-number summary. The box boundaries show Q1 and Q3, the line inside shows the median, and the whiskers extend to the minimum and maximum values—excluding outliers.
- What is a mild outlier vs. an extreme outlier?
- Mild outliers are between 1.5 and 3.0 times the IQR away from the quartiles. Extreme outliers are greater than 3.0 times the IQR away.
- Can the median be equal to Q1 or Q3?
- Yes, in datasets with many repeated values, the quartiles and the median can overlap and have identical numerical values.
- How does skewness affect the five-number summary?
- If the distance from Q1 to the Median is larger than from the Median to Q3, the data is left-skewed. If the opposite is true, it is right-skewed.