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Prealgebra / LEVEL 14 · DIFFICULTY 5/5

Data & Statistical Reasoning

Summarize actual observations, read displays and examine what summaries omit.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST 14.1Mean, Median, Mode & RangeRead the lesson
  2. MINI QUEST 14.2Tables, Charts & Weighted SummariesRead the lesson
  3. MINI QUEST 14.3What Summaries Can & Cannot Tell UsRead the lesson
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STAGE 14.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Mean, Median, Mode & Range

Useful preparation: Decimal Operations & Rounding

Goal: Choose and compute a summary while preserving its meaning.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

The mean redistributes the total equally

Add all observations and divide by their count. Each observation contributes according to its frequency, so repeated values count repeatedly.

mean = sum / count

The median uses sorted position

Sort first. For an odd count, use the middle observation. For an even count, average the two middle observations.

median depends on order, not total

Mode and range answer different questions

A mode is a most frequent value; there can be more than one. Range is maximum minus minimum and describes overall spread rather than a typical value.

range = maximum − minimum
WORKED EXAMPLE 1

Find the mean and median of 2, 8, 3, 7, 20.

  1. The sum is 40, so the mean is 8.
  2. Sorted values are 2, 3, 7, 8, 20; the middle is 7.

mean 8; median 7

WORKED EXAMPLE 2

Find the median of 4, 11, 6, 9.

  1. Sort: 4, 6, 9, 11.
  2. Average the middle two, 6 and 9.

7.5

Common pitfalls

Possible mix-up: Find a median without sorting.

The middle must be located in sorted order.

Possible mix-up: Every data set has exactly one mode.

Several values can tie for highest frequency.

Explain it to yourself

Which summary changes most when one very large value is added?

Preview the eight practice prompts
  1. Find the mean of 6, 8, 10 and 12.
  2. Find the median of 9, 2, 13, 5 and 7.
  3. Find the median of 4, 10, 7 and 15.
  4. Find the mode of 3, 5, 3, 8, 5, 3 and 9.
  5. Find the range of −4, 6, 1, −2 and 9.
  6. The mean of five numbers is 14. Four numbers sum to 51. Find the fifth.
  7. A cyclist records daily distances of 8, 12, 10 and 18 km. How far must the fifth day be for a five-day mean of 12 km? New context
  8. A weather log contains temperatures 11, 12, 13, 14 and 35. What is the median temperature? New context
Open stage 14.1 in the student workspace →

STAGE 14.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Tables, Charts & Weighted Summaries

Useful preparation: Mean, Median, Mode & Range · Percent, Part & Whole

Goal: Read scales and frequencies before calculating from a display.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Read the labels before the values

Check what each category represents, which units are shown and whether the axis starts at zero. Equal visual lengths can represent different values on different scales.

value = tick number × units per tick + axis start

A frequency counts observations

A table listing value and frequency is compressed raw data. Multiply each value by its count before finding the mean, and divide by total frequency.

weighted mean = Σ(value × frequency)/Σfrequency

Pie sectors represent parts of a whole

An angle out of 360 degrees corresponds to the same fraction of observations, provided all categories partition the whole.

category fraction = sector angle / 360°
WORKED EXAMPLE 1

A table lists score 2 for 3 students and score 5 for 2 students. Find the mean.

  1. Total score is 2(3) + 5(2) = 16.
  2. There are 5 students.

3.2

WORKED EXAMPLE 2

A 72° pie sector represents cycling in a survey of 150 people. How many chose cycling?

  1. 72/360 = 1/5 of the circle.
  2. Take 1/5 of 150.

30

Common pitfalls

Possible mix-up: Average the listed values without their frequencies.

A value with frequency ten represents ten observations.

Possible mix-up: A bar twice as tall always means twice the value.

That claim requires a common zero baseline and consistent scale.

Explain it to yourself

What must be checked before comparing the heights of bars on two different charts?

Preview the eight practice prompts
  1. A frequency table has value 1 with count 4 and value 3 with count 6. How many observations are there?
  2. In that table, value 1 occurs 4 times and value 3 occurs 6 times. Find the mean.
  3. A bar chart axis starts at 0 with ticks every 5 units. A bar reaches the seventh tick above zero. What value does it show?
  4. A pie chart category occupies 54°. What percent of the whole is it? Enter only the numerical percent.
  5. A table lists: 0 books read by 2 people, 1 book by 5 people, and 2 books by 3 people. How many books were read in total?
  6. A graph vertical axis begins at 40 and rises by 2 per tick. A point is 6 intervals above the starting level. What value is plotted?
  7. A survey of 240 students has a 105° sector for walking to school. How many students does the sector represent? New context
  8. A store sells 3 notebooks at $4 each and 7 at $6 each. What is the mean selling price per notebook, in dollars? New context
Open stage 14.2 in the student workspace →

STAGE 14.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

What Summaries Can & Cannot Tell Us

Useful preparation: Tables, Charts & Weighted Summaries

Goal: Recognize outliers, group weights and limits of inference.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Different data can share a summary

A mean gives a total per observation, but it does not determine spread or every individual value. Compare a center with the range or full distribution when possible.

2, 4, 6 and 0, 4, 8 share mean 4

Combine groups using their sizes

A group mean must be weighted by the number of observations it represents. Averaging two group means directly is valid only for equal group sizes.

combined mean = total of all values / total count

Describe the evidence actually collected

A result from a self-selected sample need not represent everyone. A relationship in data does not by itself establish a cause. Avoid inferring an individual result from a group mean.

association alone does not establish causation
WORKED EXAMPLE 1

One group of 4 has mean 10; another group of 6 has mean 15. Find the combined mean.

  1. Recover group totals 40 and 90.
  2. Divide total 130 by total count 10.

13

WORKED EXAMPLE 2

Data 5, 6, 7 have mean 6. Add the value 30. Find the new mean.

  1. The new total is 48 with 4 observations.
  2. The large value changes the mean substantially.

12

Common pitfalls

Possible mix-up: A group mean of 10 means everyone scored 10.

Many different individual scores can have that mean.

Possible mix-up: Two equal means guarantee equal spreads.

Center and spread describe different aspects of data.

Explain it to yourself

What additional information would help decide whether a mean represents a typical observation?

Preview the eight practice prompts
  1. A group of 3 has mean 8 and a group of 7 has mean 12. Find the combined mean.
  2. Do the lists 1, 5, 9 and 4, 5, 6 have the same mean? Enter yes or no.
  3. Find the range of 1, 5, 9.
  4. The data 4, 5, 6 gain a new value 25. Find the new mean.
  5. A class mean score is 80. Must every student have scored at least 80? Enter yes or no.
  6. Two equal-size groups have means 14 and 20. Find their combined mean.
  7. A survey asks only chess-club volunteers about their favorite game. Does this design guarantee a representative result for the whole school? Enter yes or no. New context
  8. A shop reports that umbrella sales and rainy days rise together. Does that observation alone prove umbrella sales cause rain? Enter yes or no. New context
Open stage 14.3 in the student workspace →
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