Measures of Center
Find the mean of these positive numbers: 2.5,3.0,4.52.5,3.0,4.5.
3.33
Is this a statistical question? “How many pets does Mrs. Lee have?” Answer: yes or no, and briefly explain why.
Not Statistical
A teacher surveys 10 students for favorite fruit and gets: apple, banana, apple, grape, apple, banana, pear, grape, apple, banana. Make a frequency table for each fruit.
Frequency table: apple: 4; banana: 3; grape: 2; pear: 1.
What type of graph would you use to show exact frequencies of a small set of decimal data: dot plot or histogram? Give one reason.
Dot plot for small exact values because it shows each individual value and repeats.
A dot plot shows student quiz scores at 7.07.0 (one dot), 8.08.0 (two dots), and 9.09.0 (three dots). What score is most common?
Most common score: 9.09.0 (three dots).
Calculate the median of the set: 1.2,3.4,2.6,4.01.2,3.4,2.6,4.0.
3
Decide if this is a statistical question: “What is the shoe size of the student who sits at desk 12?” Explain your reasoning.
Not Statistical
Given data (ages in years): 11,12,12,11,13,12,1111,12,12,11,13,12,11. Create a frequency table showing each age and its count.
Age 11: 3; Age 12: 3; Age 13: 1.
On a dot plot, how would you represent three students who each read 12.512.5 minutes? Describe the mark placement.
Place three dots stacked above the 12.512.5 mark, one above another.
If a dot plot has a cluster of dots around 3.53.5 and only one dot at 0.50.5, what does that tell you about the data’s distribution?
The data cluster near 3.53.5; 0.50.5 is an outlier or rare value.
A student has test scores of 78.5,85.0,90.5,88.0,92.078.5,85.0,90.5,88.0,92.0.
86.8
Explain why “How long does it take to run a mile?” can be a statistical question for a class, and give two pieces of data you might collect (use decimal times).
It is statistical
From the list of numbers: 2.5,3.0,2.5,4.0,3.0,3.02.5,3.0,2.5,4.0,3.0,3.0, make a frequency table and identify the mode(s).
Values: 2.5(2),3.0(3),4.0(1)2.5(2),3.0(3),4.0(1). Mode = 3.03.0.
A table shows number of books read by students: 0,1,2,2,3,3,3,40,1,2,2,3,3,3,4. Sketch (describe) a dot plot for these data (list x-values and number of dots).
Dot plot: 0:10:1 dot, 1:11:1 dot, 2:22:2 dots stacked, 3:33:3 dots stacked, 4:14:1 dot.
Given a dot plot with five dots at 2.02.0, two dots at 4.04.0, and one dot at 6.06.0, find the mean of the data set (show work).
Data expanded: five 2.02.0 gives total 5×2.0=10.05×2.0=10.0; two 4.04.0 gives 8.08.0; one 6.06.0 gives 6.06.0. Sum = 24.024.0. Mean = 24.0÷8=3.024.0÷8=3.0.
Find the median of these numbers ordered smallest to largest: 2.1,3.3,3.3,4.7,5.0,6.22.1,3.3,3.3,4.7,5.0,6.2.
4
For each of the following, say whether it is statistical and why: a) “What is the temperature today in our classroom?” b) “How many hours do students in our class sleep on a school night?”
A) Not Statistical
B) Yes
A survey recorded minutes of reading per day for 12 students: 12.5,7.0,20.0,15.5,7.0,12.5,30.0,7.0,15.5,12.5,20.0,12.512.5,7.0,20.0,15.5,7.0,12.5,30.0,7.0,15.5,12.5,20.0,12.5. Create a frequency table showing each distinct value and its frequency.
Frequencies: 7.0(3),12.5(4),15.5(2),20.0(2),30.0(1)7.0(3),12.5(4),15.5(2),20.0(2),30.0(1).
Given grouped data with intervals 0−40−4 (count 3), 5−95−9 (count 6), 10−1410−14 (count 4), describe how a histogram would look and state which interval has the highest frequency.
Histogram bars: 5−95−9 tallest (height 6), 0−40−4 shorter (3), 10−1410−14 medium (4).
A dot plot shows times (minutes) students spent on homework: 10.010.0 (2 dots), 15.015.0 (3 dots), 25.025.0 (1 dot). Find the median and explain your steps.
List data in order: 10.0,10.0,15.0,15.0,15.0,25.010.0,10.0,15.0,15.0,15.0,25.0 (6 data points). Median = average of middle two (3rd and 4th): 15.0+15.02=15.0215.0+15.0=15.0.
Explain which measure (mean or median) is better to represent the “typical” value when a data set has one very large outlier, and give a short numeric example with positive decimals to show why.
Median is more robust with outliers. Example: data 2.0,2.5,3.0,3.5,50.02.0,2.5,3.0,3.5,50.0. Mean = 61.05=12.2561.0=12.2, median = 3.03.0. Median better represents typical value.
Write a statistical question about the heights of students in the grade that would require collecting data and include the units and how you would record at least five decimal or decimal-like values (e.g., 4.9,5.14.9,5.1 feet).
Example question: “What are the heights (in feet) of students in 6th grade?” Record: 4.9,5.1,4.8,5.4,5.04.9,5.1,4.8,5.4,5.0 (units: feet).
Explain how you would group the following decimal data into class intervals for a histogram and show the frequency counts for each interval: 0.5,1.2,1.8,2.0,2.4,2.6,3.1,3.4,3.9,4.20.5,1.2,1.8,2.0,2.4,2.6,3.1,3.4,3.9,4.2. Choose intervals of width 1.01.0 (for example 0.5−1.40.5−1.4, 1.5−2.41.5−2.4, etc.) and fill counts.
Intervals and counts (example with intervals width 1.01.0 starting at 0.50.5): 0.5−1.4:10.5−1.4:1 (contains 0.5,1.20.5,1.2? note 1.2 is in next if using 1.5 lower bound—teachers adjust boundaries), Better choice (centered intervals): 0.5−1.4:10.5−1.4:1 (0.5), 1.5−2.4:21.5−2.4:2 (1.8, 2.0), 2.5−3.4:32.5−3.4:3 (2.6, 3.1, 3.4), 3.5−4.4:33.5−4.4:3
Explain one advantage and one disadvantage of using a histogram instead of a dot plot when data include many decimal values.
Advantage: histograms handle many values and show shape for grouped data. Disadvantage: they hide exact individual values and can obscure small clusters or repeated decimals.
A dot plot for 11 students has dots at: 1.01.0 (1), 2.02.0 (2), 3.03.0 (3), 4.04.0 (3), 10.010.0 (2). Compute the mean and median, then explain how the outlier (the 10.010.0 values) affects the mean compared to the median. Provide your numeric results.
Full list (with counts): 1.0(1),2.0(2),3.0(3),4.0(3),10.0(2)1.0(1),2.0(2),3.0(3),4.0(3),10.0(2) gives ordered data of 11 items. Sum = 1.0+2(2.0)+3(3.0)+4(4.0)+2(10.0)=1.0+4.0+9.0+16.0+20.0=50.01.0+2(2.0)+3(3.0)+4(4.0)+2(10.0)=1.0+4.0+9.0+16.0+20.0=50.0. Mean = 50.0÷11≈4.545…≈4.5550.0÷11≈4.545…≈4.55. Median is the 6th value (middle of 11): counts by position: 1 (1), 2–3 (2.0s), 4–6 (3.0s) → 6th value = 3.03.0. Median = 3.03.0. Effect: outliers (the two 10.010.0 values) raise the mean more than the median; median stays at 3.03.0 while mean increases to about 4.554.55.