Geometry & Statistics
Understanding Data Distributions: Center and Variability
Grade 6
- ✓By the end of this lesson students will be able to describe the center, spread, and overall shape of a data distribution.
- ✓By the end of this lesson students will be able to calculate and interpret the mean and median as measures of center for a numerical data set.
- ✓By the end of this lesson students will be able to calculate and interpret the range as a measure of variability for a numerical data set.
- ✓By the end of this lesson students will be able to identify and describe the impact of outliers on measures of center and variability.
Key concepts
A data distribution shows all the possible values (or intervals) of the data and how often they occur. When we look at a distribution, we describe its overall shape, its center, and its spread (or variability). The shape can be symmetric, skewed, or have clusters and gaps. The center tells us where the 'middle' of the data is, and the spread tells us how 'stretched out' or 'bunched up' the data is.
The mean (often called the 'average') is a measure of center found by adding all the values in a data set and then dividing by the number of values. It represents a 'fair share' if the total were distributed equally among all data points. The mean is sensitive to extreme values (outliers).
The median is the middle value in a data set when the values are arranged in order from least to greatest. If there is an odd number of data values, the median is the single middle value. If there is an even number of data values, the median is the average (mean) of the two middle values. The median is less affected by extreme values (outliers) than the mean.
The range is a measure of variability that describes the spread of a data set. It is calculated by subtracting the smallest value from the largest value in the data set. A larger range indicates greater variability, meaning the data points are more spread out. Other measures of variability include the Interquartile Range (IQR) and Mean Absolute Deviation (MAD), which you may learn about later.
An outlier is a data value that is significantly different from the other values in a data set. Outliers can be much larger or much smaller than the rest of the data. They can have a significant impact on some measures of center (like the mean) and variability (like the range).
Key facts to remember
- 1A data distribution shows how data values are spread out.
- 2Measures of center (mean, median) describe the 'typical' value in a data set.
- 3The mean is the average of all values; it is sensitive to outliers.
- 4The median is the middle value when data is ordered; it is less affected by outliers.
- 5Measures of variability (range) describe how spread out the data values are.
- 6The range is the difference between the largest and smallest values.
- 7An outlier is a data value that is much larger or smaller than the rest.
- 8Outliers can significantly change the mean and range, but usually have less impact on the median.
Worked examples
Example 1
A group of Grade 6 students recorded the number of books they read last month: 5, 8, 3, 10, 4. Find the mean, median, and range of the number of books read.
Answer
Mean = 6 books, Median = 5 books, Range = 7 books.
Always order the data first when finding the median or range.
Example 2
The scores on a math quiz for 8 students were: 75, 90, 80, 95, 70, 85, 100, 60. Find the mean, median, and range of the quiz scores.
Answer
Mean = 81.875, Median = 82.5, Range = 40.
When there's an even number of data points, the median is the average of the two middle values.
Example 3
A baker recorded the number of cakes sold each day for a week: 12, 15, 10, 13, 14, 11, 40. Identify any outlier. Then, calculate the mean, median, and range both with and without the outlier, and describe its impact.
Answer
Outlier: 40. With outlier: Mean ≈ 16.43, Median = 13, Range = 30. Without outlier: Mean = 12.5, Median = 13.5, Range = 5. The outlier significantly increased the mean and range, but had little effect on the median.
The median is often a better measure of center than the mean when a data set contains outliers.
Common mistakes
- ✗Forgetting to order the data before finding the median or range.
- ✗Incorrectly calculating the median for an even number of data values (e.g., picking one of the two middle values instead of averaging them).
- ✗Confusing the mean and median.
- ✗Not showing all steps, especially for calculations like the mean or median of an even set.
- ✗Misinterpreting what the mean, median, or range represent in the context of the problem.
Exam tips
- ★Always read the problem carefully to understand what measures you need to calculate.
- ★For median and range, always start by ordering your data from least to greatest.
- ★Show all your work for calculating the mean (sum and division) and median (identifying middle values or their average).
- ★Double-check your calculations, especially when adding many numbers or dividing.
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