Measures of Central Tendency
Indholdsfortegnelse
- The Problem
- Is Sleep Affecting Math Performance?
- Task 1
- Determining the Typical Value
- Task 2
- Measures of Central Tendency
- Further Task
- Further Tasks - Measures of Central Tendency
Is Sleep Affecting Math Performance?
Recently, the Grade 7 mathematics teacher noticed a concerning change in student performance. Section A, whose students had almost the same math grades as Section B in the previous quarter, received noticeably lower math grades this quarter. Meanwhile, students in Section B generally appear less tired and more attentive during class discussions.[br][br]The math teacher wonders whether sleep habits could be affecting students’ academic performance.
After collecting the data on daily sleeping hours, the teacher wants to:[br][br]1. Determine the typical number of hours students sleep in each section.[br][br]2. Compare the sleeping habits of Section A and Section B.[br][br]3. Analyze and interpret the results. Then, write a conclusion and recommendation based on the findings.
Determining the Typical Value
Typical Value
A [b]typical value[/b] is a number that best represents or summarizes a set of data. It gives you an idea of what is “normal” or “usual” in the group.[b][i] Instead of looking at every single number, we use one value to describe the whole set.[/i][/b][br][br]In this first task, [b]determine the typical values of daily sleeping hours for each section[/b] by studying the data representations given.
Data Representations
In the applet below, there are three types of data representations. The left side of the graphics shows the representations for Section A, and the right side shows those for Section B.[br][br]The data is displayed using a bar chart, a pie chart, and a stem-and-leaf plot. You can click the check boxes in the applet to display the corresponding data representations for each data set.
Section A
What is the [b]typical daily sleeping hours[/b] of Section [b]A[/b]?
Explain or justify your answer about the typical daily sleeping hours of Section A.
Section B
What is the[b] typical daily sleeping hours [/b]of Section [b]B[/b]?
Explain or justify your answer about the typical daily sleeping hours of Section B.
Measures of Central Tendency
In statistics, when we want to describe what is typical or usual in a set of data, we use special numbers called [b][i]measures of central tendency[/i][/b].[br][br]“Central” means middle, and “tendency” means what usually happens.[br][br]So, measures of central tendency are numbers that tell us what the [b]center or typical value of a data set [/b]is.[br][br]There are three main measures:[br][list=1][*]Mean[/*][*]Median[/*][*]Mode[/*][/list]
Mean
The [b]mean[/b] is what most students call the [b]average[/b].
Median
The [b]median[/b] is the [b]middle number [/b]when the data is arranged from smallest to largest.
Mode
The [b]mode[/b] is the number that appears [b]most often or most common value[/b].
Second Task
In the second task, answer the following questions about the measures of central tendency (mean, median, and mode) of the data set from the first task. Then, answer the questions that ask you to interpret the results and write a conclusion about the problem.
Measures of Central Tendency of Sleeping Hours of Sections A and B.
The applet below allows users to display the [b]computed measures of central tendency [/b]for each section by [b][i]clicking the corresponding button[/i][/b].[br][br]You may also use the [b][i]checkboxes[/i][/b] to show or hide how each measure (mean, median, and mode) is [b]represented in the bar charts for both sections.[/b]
Deciding Measure of Central Tendency
Choose the measure of central tendency that best represents the typical daily sleeping hours of Section A and Section B.
Explaining and justifying
Explain and/or justify why your chosen measure is appropriate.
Representativeness of the Chosen Measure
For which section does the chosen measure of central tendency better represent the entire data set?
Explaining and justifying
Explain and/or justify your answer.
Interpreting data
Which section has a higher typical number of sleeping hours based on your chosen measure of central tendency? What does this suggest?
Interpreting data
Based on your analysis, do you think sleep may be one possible reason why Section A's math performance decreased this quarter? Explain your answer using statistical evidence.
Communicating data
If you were the math teacher, what recommendation would you give to the students based on the results of the study?
Further Tasks - Measures of Central Tendency
Measures of Central Tendency and Probability
What is the [b]chance of a student[/b] in Section A to have [b]7 hours of sleep daily? 9 hours? 2 hours? [/b]
Measures of Central Tendency and Probability
What is the [b]chance of a student[/b] in Section B to [b]have 7 hours of sleep daily? 9 hours? 2 hours? [/b]
Effects of Data Transformation upon Mean
What happens to the [b]MEAN[/b] of each section if all students in a section [b][i]get an extra x number of hours of sleep[/i][/b]?[br][br]
Effects of Data Transformation upon Median
What happens to the [b]MEDIAN[/b] of each section if all students in a section [b][i]get an extra x number of hours of sleep[/i][/b]?[br][br]
Effects of Data Transformation upon Mode
What happens to the [b]MODE[/b] of each section if all students in a section [b][i]get an extra x number of hours of sleep[/i][/b]?[br][br]