Statistical questions and summarizing data
Recognizing a question that expects varied answers, and summarizing a numerical data set with center, spread and shape
About three to four lessons of 45 to 60 minutes
'How old is the school principal?' versus 'How old are the students in Grade 6?'
The first question has one fixed answer, whatever it is -- ask it ten times and get the same number every time. The second question expects DIFFERENT answers from different students, because ages genuinely vary across a class. Only the second is a statistical question, and spotting that difference is the first skill in this unit.
Once you have real, varying data, the second skill is summarizing it fairly: not just one number, but a small set of numbers that together describe how many data points there are, where the data is centred, how spread out it is, and what shape it takes.
- 'What is the capital of France?'not statistical -- one fixed answer, no variability
- 'How many pets does each student in class 5B have?'statistical -- answers vary from student to student
- Ages: 10, 11, 10, 12, 11, 10, 137 observations, mean 11, median 11, mode 10, range 3
- A dot plot of the same agesshows the shape: clustered around 10-11, with one higher outlier at 13
What students will be able to do
Students will recognize a statistical question as one that anticipates variability in the data collected to answer it, and will summarize a numerical data set by reporting the number of observations, a measure of centre (mean, median, or mode), a measure of spread (range), and a description of the data's overall shape.
- I can explain why a statistical question expects varied answers, while a non-statistical question does not.
- I can write a statistical question about a real topic.
- I can report the number of observations in a data set.
- I can compute the mean, median and mode of a numerical data set.
- I can compute the range and describe the overall shape of a data set.
Standards this unit teaches
- 6.SP.A.1Common Core (US)Recognize statistical questions
Recognize a statistical question as one that anticipates variability in the data that answers it.
- 6.SP.B.5Common Core (US)Summarize data sets
Summarize numerical data by reporting the number of observations, centre, spread, and overall shape.
- AC9M6ST02Australian Curriculum v9 (ACARA)Represent varied data (Year 6)
Acquire, validate and represent data for categories and numbers using spreadsheets and report on the data distribution.
- AC9M7ST01Australian Curriculum v9 (ACARA)Mean, median, mode and range (Year 7)
Collect discrete and continuous numerical data, calculate the range, mean, median and mode, and decide which measure best describes the data. Australia's fullest formal treatment of the numerical summary measures used here sits at Year 7, so that half of this unit runs about a year ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Statistical question
- a question that anticipates variability, expecting different answers from different data points
- Data
- the collected answers or measurements gathered to answer a question
- Mean
- the total of all values divided by how many values there are, the fair-share average
- Median
- the middle value once the data is placed in order
- Mode
- the value that appears most often in a data set
- Range
- the difference between the greatest and least value in a data set
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Does the question expect varied answers?
ConcreteTest any question with one simple check: if you asked it about many different people or things, would you expect the answers to be the same every time, or would they genuinely vary? 'How old is my teacher?' has one fixed true answer -- not statistical. 'How old are the students in my class?' will genuinely differ from student to student -- statistical.
This check matters because it decides whether collecting data even makes sense. There is no point collecting many answers to a question with only one true answer, but a statistical question needs a whole data set, which is exactly what the rest of this unit summarizes.
- Is 'How many pets does each student in class 5B have?' a statistical question? Why?
- Is 'What is the capital of France?' a statistical question? Why?
2. Measures of centre: mean, median, mode
PictorialOnce you have varied data, a single 'centre' number summarizes it. The mean shares the total out equally: add every value, then divide by how many values there are. The median is the middle value once the data is sorted in order. The mode is the value that shows up most often.
For the ages 10, 11, 10, 12, 11, 10, 13 (7 observations): the mean is (10+11+10+12+11+10+13) divided by 7, which is 77 divided by 7, equalling 11. Sorted, the data is 10, 10, 10, 11, 11, 12, 13, so the middle (4th of 7) value, the median, is also 11. The most frequent value, the mode, is 10, appearing three times.
Find the mean, median and mode of these quiz scores: 4, 6, 4, 8, 6, 4.
- Mean: add all values, 4+6+4+8+6+4 = 32, then divide by 6 observations: 32 / 6 = 5.33 (rounded to two decimal places).
- Median: sort the data: 4, 4, 4, 6, 6, 8. With 6 values (even), average the two middle values (3rd and 4th): (4 + 6) / 2 = 5.
- Mode: 4 appears three times, more than any other value.
Answer: Mean is about 5.33, median is 5, and mode is 4.
- How do you find the median when there is an even number of values?
- Can a data set have more than one mode, or no mode at all? Explain.
3. Reporting spread and shape
AbstractCentre alone does not fully describe a data set -- two very different-looking sets of data can share the same mean. The range, the greatest value minus the least value, measures how spread out the data is. Describing the overall shape (clustered tightly, spread evenly, or with an outlier standing apart) rounds out a full summary.
For the quiz scores 7, 9, 8, 10, 9, 7, 9, 8 (8 observations): mean is 67 divided by 8, which is 8.375. Sorted, 7, 7, 8, 8, 9, 9, 9, 10, so the median (average of the 4th and 5th values) is (8 + 9) / 2 = 8.5. Mode is 9 (appears three times). Range is 10 - 7 = 3. A dot plot shows the scores clustered tightly between 7 and 10, with no outliers.
Report the number of observations, centre, and spread for: 12, 15, 12, 18, 13.
- Number of observations: 5.
- Mean: (12+15+12+18+13) / 5 = 70 / 5 = 14.
- Median: sorted 12, 12, 13, 15, 18, the middle (3rd of 5) value is 13.
- Mode: 12 (appears twice).
- Range: 18 - 12 = 6.
Answer: 5 observations. Mean 14, median 13, mode 12. Range 6.
- Why does reporting only the mean give an incomplete picture of a data set?
- What does a small range tell you about how spread out the data is, compared to a large range?
Common misconceptions and how to address them
MisconceptionAny question involving numbers is automatically a statistical question.
Why it happens: Students associate 'statistics' with 'numbers' rather than with the specific idea of expected variability in the answers.
How to address it: The number check is not what matters -- the variability check is. 'How many days are in a week?' involves a number but has one fixed answer, so it is not statistical. 'How many hours did each student sleep last night?' expects varied numeric answers, so it is.
MisconceptionThe average of a data set always means the mean, and median and mode are lesser, optional measures.
Why it happens: Everyday language uses 'average' loosely, almost always meaning the mean, so students assume it is the only measure worth reporting.
How to address it: Mean, median and mode are all legitimate centre measures, and a complete summary reports the ones that are informative for the data. Show a data set with an extreme outlier where the mean is pulled far from where most of the data actually sits, but the median stays representative -- that is exactly why more than one centre measure matters.
MisconceptionOnce the mean is reported, the data set is fully summarized.
Why it happens: Students treat 'summarize the data' as a single-number task rather than a small set of complementary numbers.
How to address it: A complete summary, as this standard requires, reports the number of observations, a centre measure, a spread measure (the range), and the overall shape. Two data sets can share the exact same mean but look completely different once their range and shape are compared.
Guided practice (with answers)
1. Is 'How many hours did each student in the class sleep last night?' a statistical question?
Answer: Yes, because the answers genuinely vary from student to student.
2. Find the mean, median and mode of: 4, 6, 4, 8, 6, 4.
Answer: Mean is about 5.33 (32 / 6), median is 5 (average of 4 and 6), mode is 4.
3. Find the range of: 4, 6, 4, 8, 6, 4.
Answer: 4, since 8 - 4 = 4.
4. How many observations are in the data set 12, 15, 12, 18, 13?
Answer: 5 observations.
5. Find the median of: 12, 15, 12, 18, 13.
Answer: 13, the middle value once sorted: 12, 12, 13, 15, 18.
Independent practice worksheets
Set the matching ChalkBee statistics worksheets for independent work. The answer keys are computed in code, so they are never wrong. Pair the computation practice with writing and sorting statistical versus non-statistical questions, since this standard is about reasoning as much as computing.
Differentiation
- Sort a set of prepared questions into 'statistical' and 'not statistical' piles as a hands-on classifying activity before writing original questions.
- Use small data sets (5 or 6 values) with whole-number, exact means at first, before introducing sets whose mean requires rounding.
- Provide a fixed checklist for a full summary (count, mean, median, mode, range) so no piece is forgotten.
- Use the dot plot figure for every data set at first, so 'shape' has something visible to point to before describing it in words.
- Compare two data sets with the same mean but very different ranges, and discuss what the range reveals that the mean alone does not.
- Collect a small real data set from the class (such as the number of siblings each student has) and produce a full written summary.
- Investigate how a single extreme outlier changes the mean much more than it changes the median, using a data set of the student's own construction.
- Preview a box plot (also built from the same data) as a second way to show spread and shape, once a dot plot is comfortable.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling statistical-question recognition and a full summary.
1. Is 'What is the capital of France?' a statistical question? Why or why not?
Answer: No, because it has one fixed correct answer and does not anticipate variability.
2. Find the mean, median and mode of: 12, 15, 12, 18, 13.
Answer: Mean is 14 (70 / 5), median is 13 (the middle sorted value), mode is 12.
3. Find the range of: 12, 15, 12, 18, 13.
Answer: 6, since 18 - 12 = 6.
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 recognizing statistical questions (section 1), Lesson 2 measures of centre (section 2), Lesson 3 spread and shape plus a full summary (section 3), Lesson 4 (optional) a small real class data collection and write-up plus the exit ticket.
- Language to keep saying: does the question expect varied answers, a full summary needs count plus centre plus spread plus shape, mean is not the only average that matters. These target the three main misconceptions directly.
- This unit assumes fluent whole-number division (for the mean) and comfortable ordering of a small list of numbers (for the median); if either is shaky, a quick warm-up pays off before this unit starts.
- Curriculum note: ACARA v9's Year 6 statistics descriptor (AC9M6ST02) covers acquiring and representing varied data and reporting on its distribution, a partial match to this unit. Australia's fullest formal treatment of mean, median, mode and range together, deciding which measure best describes a data set, sits at Year 7 (AC9M7ST01), so that computational half of this unit runs about a year ahead of the ACARA placement.
- Present mode and print both work: use Present to sort statistical versus non-statistical questions live with the class and build a dot plot together, then print for independent summarizing practice.