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   Introducing
Distribution |
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| Â Â Â Â Lesson
2: Inventing Statistics - Center |
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Overview of the Lesson / Preparing for the Lesson / Lesson Activities / Assessment
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| Â Â Â Â Students'
Ways of Thinking |
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Students' thinking about how to measure the center or central tendency in a measurement data generally falls into three different types of reasoning: (a) Convention, (b) Repeated Values, (c) center clump, and (d) mid-range. How students think about what to measure about central tendency is influenced by the display of the data. For example, a pair of fifth-grade students invented the display. It facilitates thinking about the clump of data in the center.
Notice that in this display, the data seem to form a hill located approximately at the mid-point of the range of values.
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Convention.
Some students have been taught that if they have a batch of data, they should calculate a statistic, usually the mean. They are typically unaware of any of the properties of the mean, so we usually alter instruction to make some of the qualities of the mean more visible. For example, students investigate the effects of extreme values on the mean. Students can also explore how means represent a ‘per case' point-of-view.
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Repeated Values.
Many students reason that if two or more people agree about a measured value, then that value is more likely the “right” one. Hence, they suggest that the mode is the best guess of the true measure. However, some students look for repeated values even when the data are multimodal. For example, the method proposed by one team of fifth-grade students, displayed below, does not choose the modal value. It asks instead that users choose the most “reasonable” value.
1. Sort into groups of ten
2. Find the 150s' group
3. Order/ sort from least to greatest in 150s' group
4. Identify doubles (two 152s and two 158s)
5. 152 cm is the best guess because it is reasonable. |
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Center Clump.
The majority of students who do not have a prior orientation to the mean invent analogs to the median. Their reasoning is guided by the appearance of a center clump in the data (see earlier display), when the data are grouped and ordered. Students are attracted to the relative frequencies of the values in the data, likely because these values literally occupy more space (e.g., they are higher). Most solutions involve finding the middle value of this center bin, a workable solution for measurement data. However, some students independently invent the median, guided by a sense of middle as splitting the data into two parts of equal count. For example, a pair of fifth grade students came up with the following method:
1. Data out (the data were on cards, one value per card)
2. Grouping 2 nd digit (tens) and order from least to greatest in groups
3. Put from least to greatest of a group
4. Count all cards, and then divide the total by 2
5. Count half of the total from least to greatest – first number
6. Count half of the total from greatest to least – second number
7. Find the middle value between the first and second numbers |
Classmates objected when the median value was not instantiated by an actual measurement, but were persuaded by appeal to the measurement process: The median represented a value that might have easily been someone's actual measurement. It was a “possible measurement.” This form of student reasoning signaled a shift away from considering only cases toward considering the aggregate.
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Mid-Range.
Some students have a distance-based sense of center. They find the difference between the least and greatest value, and then find the mid-point of that distance. For example, one student's method was:
1. Subtract from the greatest to least
2. Find ½ of the difference
3. Add ½ of the difference to the lowest number |
This idea is fine but may not work so well given sample-to-sample variation. For example, one very extreme measurement might shift the value outside of the center clump. A teacher might want to ask students to try out an example with one “wild measurer” and the rest not.
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Last Updated:
April 14, 2006
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