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250- 300 WORDS ONLY Focus on what you learned that made an impression, what may have surprised you, and what you found particularly beneficial and why. Specifically:

• What did you find that was really useful, or that challenged your thinking?
• What are you still mulling over?
• Was there anything that you may take back to your classroom?
• Is there anything you would like to have clarified?

THE Weekly Reflection will be graded on the following criteria for a total of 5 points:

• Reflection is written in a clear and concise manner, making meaningful connections to the investigations & objectives of the week.
• Reflection demonstrates the ability to push beyond the scope of the course, connecting to prior learning or experiences, questioning personal preconceptions or assumptions, and/or defining new modes of thinking.
• NOTES TO USE: WHAT WE COVERED THIS WEEK
• Introduction & Goals

The spread, or the variability, of a distribution is measured in many ways. In this week’s investigations we will look in great detail at three measures of the spread of a distribution: IQR, Mean Average Deviation (MAD), and Standard Deviation (SD). These measures provide valuable tools for comparing distributions and evaluating the significance of their central measures.

Goals:

• Develop an understanding of mean as an indicator of fair allocation and as the “balancing point” of a set of data
• Explore deviations of data values from the mean
• Quantify variation in a distribution by calculating the Range, Interquartile Range (IQR), the Mean Absolute Deviation (MAD), or the Standard Deviation (SD) of the distribution
• Define outliers of data in terms of IQR (interquartile range) and number of deviations from the mean
• Spread refers to the variation of the distribution for a variable. You already know two simple measures of spread:
• Range: The simplest measure of spread is the range: the difference between the maximum and minimum values of the distribution. In essence, the range tells us the width of the distribution – how many different values the variable takes on, but it does not tell us how the data falls within that range. For instance, consider a class where every student scores 80% on a test, except for one student who scores 10%. This class’s scores are not very variable – nearly every student scored 80%; yet the range would be 70%, suggesting the data is more variable than it really is.
• Interquartile Range (IQR): The IQR was introduced in Week 3. It is the difference between the third quartile and the first quartile. The IQR measures the width of the middle 50% of the distribution – how different the middle values are from each other. Another way of thinking about it is as a measure of variation around the median. The smaller the IQR, the better the median represents the middle 50% of the data.

IQR is a little more focused than the range, in that it filters out the extremes of the data (the highs and the lows). However, in doing so, it ignores 50% of the data. Smaller IQRs imply less variation in the distribution. However, it is possible for the upper and lower 25% of the data to be wide-spread. For instance, consider a class where 75% of the students score 90% on a test and the remaining 25% of the students score 50%. In this case, the IQR is 0, suggesting little variation in the distribution of the scores from the median score of 90%; yet 25% of the students did not pass this test.

The appeal of both of these measures is their simplicity. Each is a simple difference of values and can be easily estimated visually on a histogram or box plot. They serve as good initial measures for comparison, used in conjunction with visual comparisons of the key features of the distribution.

In Activity A, we look more intuitively at the spread in DoW #4, and in Activity B, we introduce “deviations from the mean” as another way of measuring spread.

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