Describe the shape beyond one average
Which summaries change when a tail observation grows?
Entering midway? Sort observations before finding a median or quantile. A distribution describes how values are spread, not only their center.
Start with a concrete question
Eight original synthetic values have a controllable largest observation and a common shift. Inspect quartiles and empirical cumulative points.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict which summaries change when every observation gains the same amount.
Change one thing. Inspect what happens.
- Compute median and median-of-halves quartiles for the initial sorted values.
- Increase only the tail and compare mean, median and IQR.
- Shift all values and explain why center moves while IQR stays unchanged.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Empirical distribution: ordered observations
Horizontal: Observation value. Vertical: Cumulative fraction.
- 1. Empirical cumulative points (not interpolated) (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Observation value | Cumulative fraction |
|---|---|---|
| Empirical cumulative points (not interpolated) | 2 | 0.125 |
| Empirical cumulative points (not interpolated) | 3 | 0.25 |
| Empirical cumulative points (not interpolated) | 4 | 0.375 |
| Empirical cumulative points (not interpolated) | 5 | 0.5 |
| Empirical cumulative points (not interpolated) | 6 | 0.625 |
| Empirical cumulative points (not interpolated) | 7 | 0.75 |
| Empirical cumulative points (not interpolated) | 8 | 0.875 |
| Empirical cumulative points (not interpolated) | 20 | 1 |
Follow the calculation
- Sorted original synthetic observations: 2, 3, 4, 5, 6, 7, 8, 20.
- Median = (5+6)/2 = 5.5.
- Median-of-halves convention: Q1=(3+4)/2=3.5; Q3=(7+8)/2=7.5.
- IQR = Q3−Q1 = 4; mean = 6.875.
Describe center, spread and tail behavior together. Quartile conventions differ; this lab states its median-of-halves rule explicitly.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Describe the shape beyond one average
Recap for entering here
Sort observations before finding a median or quantile. A distribution describes how values are spread, not only their center.
The concrete question
Eight original synthetic values have a controllable largest observation and a common shift. Inspect quartiles and empirical cumulative points.
Predict which summaries change when every observation gains the same amount.
Work through the mechanism
Quartiles describe positions in a sorted distribution. This lab uses medians of the lower and upper four observations; software may use other conventions.
For [2,3,4,5,6,7,8,20], median=(5+6)/2=5.5, Q1=(3+4)/2=3.5 and Q3=(7+8)/2=7.5, so IQR=4.
Empirical cumulative points show the fraction of observations at or below each ordered position. They do not assume a smooth population density between values.
Formula and symbols
Median = average of the two central values for this even sample. IQR = Q3−Q1, with Q1/Q3 defined as medians of the halves.
Test the idea
Compute median and median-of-halves quartiles for the initial sorted values.
Increase only the tail and compare mean, median and IQR.
Shift all values and explain why center moves while IQR stays unchanged.
Assumptions and boundaries
Eight authored observations are not an independent random sample from a real population.
Tied observations produce repeated x positions; an exact empirical CDF jumps by their combined mass.
Extend your investigation
High school: Describe a fictional distribution using center, spread and unusual observations.
College: Compare quantile interpolation conventions and empirical versus population distributions.
Working professionals: Write a dashboard specification that includes tails and missingness rather than only an average.
Try a fresh case
Using median-of-halves quartiles for [1,2,3,4,5,6,7,8], what is the IQR?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Describe a fictional distribution using center, spread and unusual observations.
College investigation
Compare quantile interpolation conventions and empirical versus population distributions.
Workplace investigation
Write a dashboard specification that includes tails and missingness rather than only an average.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.