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Data Distribution
Characteristics

Learn how to describe data using its center, spread, and overall shape: the three pillars that reveal the story behind any data set.

6th Grade Math

What Is a Distribution?

When you collect data to answer a question, the values don't all land on the same number. They spread out in a pattern. That pattern is called the distribution: it shows which values occur, how often they occur, and how they're arranged.

Three Features: Center, Spread, Shape

Center
A single "typical" value that represents the middle of the data, like the mean or median.
Spread
How far apart the values are from each other, measured by range or interquartile range.
Shape
The overall pattern when data is graphed: symmetric, skewed left, or skewed right.

Why Does It Matter?

Imagine two classrooms both have a mean test score of \( 80 \). Are they the same? Not necessarily. One class might have everyone scoring between \( 75 \) and \( 85 \), while the other has scores from \( 40 \) to \( 100 \). The center is the same, but the spread is very different.

Measuring Center: Mean and Median

Mean
Add all values and divide by how many there are: \( \text{Mean} = \frac{\text{sum of values}}{n} \)
Median
The middle value when data is ordered from least to greatest. If \( n \) is even, average the two middle values.

Measuring Spread: Range and IQR

Range
\( \text{Range} = \text{max} - \text{min} \). Quick but affected by outliers.
IQR
\( \text{IQR} = Q_3 - Q_1 \). Measures the spread of the middle 50% of data: more resistant to outliers.

Describing Shape

Symmetric Skewed Right Skewed Left

The Distribution Rule

A complete description of a distribution mentions all three. Leaving one out gives an incomplete picture.

How Shape Affects Center

Symmetric Mean = Median Skewed Right Median Mean

Mean vs. Median: When to Use Which

Mean Median
Best for Symmetric data Skewed data
Affected by outliers? Yes No
Uses all values? Yes Only position matters
Example Test scores (no outliers) Home prices (extreme values)

How to Describe a Distribution

1
Identify the shape. Is the data symmetric, skewed right, or skewed left? Look at a dot plot or histogram.
2
Find the center. Calculate the mean and median. Choose the one that best fits the shape.
3
Measure the spread. Calculate the range and, if possible, the IQR to show how spread out the data is.
4
Note any outliers or gaps. Mention any unusual features such as clusters, gaps, or extreme values.

Example 1: Student Sleep Hours

A class recorded how many hours they slept last night:
\( \{6, 7, 7, 7, 8, 8, 8, 8, 9, 9\} \)

6 7 8 9 median = 8

Example 2: Weekly Allowance

Ten students reported their weekly allowance in dollars:
\( \{5, 5, 10, 10, 10, 10, 15, 15, 20, 50\} \)

$5 $10 $15 $20 $50 2 4 2 1 1 outlier

Example 3: Comparing Two Classes

Class A: Scores 70 75 80 85 90 Range = 10 Class B: Scores 70 75 80 85 90 Range = 20

Common Mistakes to Avoid

✕ Only Reporting the Mean
The mean alone does not describe the full distribution. Always mention shape and spread too.
✕ Ignoring Outliers
Outliers can dramatically change the mean. Check for extreme values before choosing your center measure.
✕ Confusing Skew Direction
"Skewed right" means the tail goes right, not the peak. The peak is on the left side of the graph.

Summary of Distribution Features

Feature What It Tells You How to Measure
Center A typical value Mean or Median
Spread How much values vary Range or IQR
Shape The overall pattern Symmetric, Skewed L/R
Outliers Unusual values Visual inspection, IQR rule

Question 1 of 5

Which of the following best describes the center of a data distribution?

Question 2 of 5

A data set has values: \( \{2, 3, 3, 4, 4, 4, 5, 5, 20\} \). Which measure of center better represents the typical value?

Question 3 of 5

A histogram shows data with a peak on the right side and a long tail stretching to the left. This distribution is:

Question 4 of 5

Two data sets both have a mean of \( 50 \). Data Set A has a range of \( 10 \) and Data Set B has a range of \( 40 \). What can you conclude?

Question 5 of 5

A student describes a data set by saying: "The average is \( 72 \)." Is this a complete description of the distribution?