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Math: Statistics and Probability

Averages, data displays, sampling and probability models: the Grade 7 statistics strand, which turns up on Grade 8 and Grade 9 forms alike.

24 topics · 94 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Mean, Median, Mode and Range

  1. How it worksHow do you find the median of a list with an even count of values instead of an odd one?
  2. How it worksWalk me through computing mean, median, mode and range for the set 4, 7, 7, 9, 12 and 15.
  3. Worked exampleFind the mean, median, mode and range of the quiz scores 88, 92, 80, 92, 60, 84 and 92.
  4. Common mistakeWhy isn't the median simply the middle number of the list exactly as it was handed to you?
  5. Going furtherWhat happens to the mean and to the median when one huge value joins a small set?
Your turneasy1 of 3
Find the mean of the quiz scores 88, 92, 80, 92, 60, 84 and 91.

Choosing the Right Measure of Center

  1. How it worksHow does the shape of a distribution decide whether mean or median describes it honestly?
  2. Worked exampleFind which average best describes the salaries 32000, 35000, 36000, 38000 and 400000.
  3. Common mistakeWhy doesn't the mode make any sense as a center for a list where every value appears once?
  4. Going furtherWhat happens to the mean and the median of a data set when every value is raised by 10?
Your turnmedium1 of 3
Five salaries are 32000, 35000, 36000, 38000 and 508925 dollars. Find the mean salary, in dollars.

Working Backwards From an Average

  1. How it worksHow does turning a mean back into a total unlock every missing-value question at once?
  2. Worked exampleFind the score needed on a sixth test to lift an average of 82 across five tests up to 85.
  3. Worked exampleFind the fourth number when four values average 19 and three of them are 12, 24 and 15.
  4. Common mistakeWhy doesn't averaging two averages give the average of the whole combined group?
Your turnmedium1 of 3
7 tests average 80 points. Write an expression, in terms of t, for the score needed on one more test so that all 7 plus 1 tests average t.

Combining Groups and Weighted Averages

  1. How it worksHow do totals rather than averages give the right answer when two unequal groups combine?
  2. Worked exampleFind the combined average when 20 students average 78 and 30 students average 88 on one test.
  3. Going furtherWhat happens to a combined average as one of the two groups grows far larger than the other?
Your turneasy1 of 3
Twenty students average 78 on a test and thirty students average 90. Find the combined average of all fifty.

Dot Plots

  1. How it worksHow do you pull the mode, the range and the gaps straight off a stack of dots with no arithmetic?
  2. Worked exampleFind the median and range of a dot plot holding 3 dots at 2, 5 dots at 3, 1 dot at 4 and 2 dots at 9.
  3. Common mistakeWhy doesn't the tallest stack of dots always sit where the mean of the data lands?
Your turneasy1 of 3
A dot plot holds 3 dots at 2, 5 dots at 3, 1 dot at 4 and 2 dots at 16, where 16 is the largest value. Find the range.

Histograms

  1. How it worksHow do the bin width and the bin edges change the picture a histogram gives of one data set?
  2. Worked exampleFind how many of 40 runners beat 30 minutes when bins of ten hold 6, 14, 12 and 8 runners.
  3. Common mistakeWhy isn't a histogram simply a bar graph with the bars pushed up against each other?
  4. Going furtherWhat happens to the shape of a histogram when the bin width is doubled?
Your turneasy1 of 3
A histogram of 40 runners uses bins of ten minutes: 0 to 10, 10 to 20, 20 to 30 and 30 to 40. The bin heights are 6, 17, 12 and 8. How many runners finished in under 30 minutes?

Bar Graphs and Line Graphs

  1. How it worksHow does a line graph turn time into a direction so that rising and falling become visible?
  2. Common mistakeWhy isn't a bar graph with a chopped-off vertical axis telling the truth about the differences?
  3. Worked exampleFind the month with the biggest jump in sales of 120, 150, 145, 200, 260 and 255 over six months.
  4. Going furtherWhat happens to a line graph's story when the spacing along the time axis is uneven?
Your turneasy1 of 3
Monthly sales run 120, 150, 145, 200, 279 and 255. Find the increase from the fourth month to the fifth.

Circle Graphs

  1. How it worksHow does a percentage turn into the exact number of degrees for its slice of the circle?
  2. How it worksHow do you get a raw count back out of a pie chart once you know the total it was built from?
  3. Worked exampleFind the degrees and the count for a 35 percent slice of a pie chart built from 480 survey answers.
  4. Common mistakeWhy doesn't a bigger slice in one pie chart mean a bigger number than a smaller slice in another?
Your turneasy1 of 3
A circle graph slice represents 45 percent of the whole. How many degrees does that slice cover?

Reading Tables and Translating Between Displays

  1. How it worksHow do you rebuild a dot plot out of a frequency table without losing a single value?
  2. Worked exampleFind the bin heights for the values 3, 7, 8, 11, 12, 13, 14, 19, 21 and 22 grouped in fives.
  3. Worked exampleFind the pie chart percentages for a tally of 12 walkers, 18 bikers, 6 drivers and 24 bus riders.
  4. Going furtherWhat happens to your reading of a data set when you swap a box plot for a dot plot?
Your turneasy1 of 3
A tally records 12 walkers, 18 bikers, 6 drivers and 32 bus riders. What percentage of the group are bus riders?

Box Plots and the Five-Number Summary

  1. How it worksHow do you pull the minimum, the quartiles, the median and the maximum out of a sorted list?
  2. How it worksHow does each of the four parts of a box plot end up holding about a quarter of the values?
  3. Worked exampleFind the five-number summary of 4, 6, 7, 9, 11, 12, 14, 18, 21, 25, 30 and 31.
  4. Common mistakeWhy isn't a long whisker a sign that more data points are living out at that end?
  5. Going furtherWhat happens to the box itself when the middle half of the data spreads much wider?
Your turneasy1 of 3
Find the range of the sorted set 4, 6, 7, 9, 11, 12, 14, 18, 21, 25, 30 and 51, where 51 is the largest value.

Interquartile Range and Outliers

  1. How it worksHow does the rule of one and a half times the IQR draw a fence around what counts as normal?
  2. Worked exampleFind the interquartile range and any outliers in 12, 14, 15, 16, 18, 19, 21, 22, 24 and 48.
  3. Common mistakeWhy isn't every value sitting far from the mean automatically an outlier?
  4. Going furtherWhat happens to the interquartile range when one extreme value is deleted from the set?
Your turnmedium1 of 3
Find the interquartile range of the sorted set 12, 14, 15, 16, 18, 19, 21, 22, 24 and 48, where 15 lies between 14 and 16.

Shape, Center and Spread

  1. How it worksHow do clusters, peaks, gaps and symmetry each show up as a feature of a distribution's shape?
  2. Worked exampleFind the center and the spread of 1, 2, 2, 3, 3, 3, 4, 4, 5 and compare them with 1, 1, 1, 5, 9, 9, 9.
  3. Going furtherWhat happens to the gap between mean and median when a distribution leans out to the right?
Your turneasy1 of 3
Find the mean of the set 1, 2, 2, 3, 3, 3, 4, 4 and 15.

Sampling and Representative Samples

  1. How it worksHow does a random draw keep every part of a population in the running for the sample?
  2. How it worksHow does a sampling frame that misses a whole group bend the result before anyone answers?
  3. Worked exampleFind the estimated left-handers in a school of 1200 when 9 of a random 75 students are left-handed.
  4. Common mistakeWhy isn't a poll of 5000 people from one website better than a random sample of only 500?
  5. Going furtherWhat happens to the spread of estimates as the sample size climbs from 20 up to 200?
Your turneasy1 of 3
In a random sample of 88 students, 12 are left-handed. Write an expression, in terms of N, for the estimated number of left-handed students in a school of N.

Biased Sampling and Convenience Samples

  1. How it worksHow do convenience samples, volunteer polls and leading questions each tilt a result a different way?
  2. Common mistakeWhy doesn't an enormous sample size ever repair a sample that was collected the wrong way?
  3. Going furtherWhat happens to a poll when the people who refuse to answer differ from the people who do?
Your turneasy1 of 3
A survey is mailed to 2241 households and 40 percent of them reply. How many households replied?

Using a Sample to Predict a Population

  1. How it worksHow does a sample proportion turn into a population estimate with a single multiplication?
  2. Worked exampleFind the estimated defective units in a shipment of 4500 when 7 of a random 150 are defective.
  3. Worked exampleFind the fish in a lake when 80 tagged fish are released and 12 of a later catch of 300 are tagged.
  4. Common mistakeWhy doesn't a single sample give an exact answer even when it was drawn perfectly at random?
Your turneasy1 of 3
In a random sample of 150 units, 7 are defective. Estimate the number of defective units in a shipment of 5089.

Comparing Two Populations

  1. How it worksHow does the overlap between two distributions measure how different the groups really are?
  2. Worked exampleFind which of two teams is truly taller from medians of 68 and 71 inches with spreads of 3 and 9.
  3. Common mistakeWhy isn't a higher mean on its own enough to claim one group beats another?
  4. Going furtherWhat happens to your conclusion when two groups share a median but have very different spreads?
Your turneasy1 of 3
One team has a median height of 68 inches and another has a median of 77 inches. Find the difference between the two medians, in inches.

Probability of a Simple Event

  1. How it worksHow does counting favorable outcomes over total outcomes build a probability from nothing?
  2. Worked exampleFind the chance of drawing a red marble from a bag holding 5 red, 7 blue and 8 green marbles.
  3. Going furtherWhat happens to that counting rule when the outcomes are not all equally likely?
  4. Common mistakeWhy isn't a probability of one half a promise of one success in every two tries?
Your turneasy1 of 3
A bag holds 5 red marbles, 7 blue marbles and 15 green marbles. Find the probability of drawing a red marble.

Complements and At-Least Problems

  1. How it worksHow does subtracting from 1 collapse a long list of cases into a single quick calculation?
  2. Worked exampleFind the chance of at least one head in three coin flips by counting the ways to get none.
  3. Common mistakeWhy isn't the complement of rolling a 5 just rolling a 1, a 2, a 3 or a 4?
Your turneasy1 of 3
A fair coin is flipped 6 times. Find the probability of getting at least one head.

Compound Events and Sample Spaces

  1. How it worksHow does an organized list or a sample space table catch every outcome exactly once?
  2. How it worksHow do you decide whether the counts for two stages should be multiplied or added?
  3. Worked exampleFind the chance two rolled number cubes total 8 by laying out the full sample space table.
  4. Worked exampleFind how many outfits come from 4 shirts, 3 pants and 2 pairs of shoes, and the chance of one.
  5. Common mistakeWhy doesn't the chance of two events both happening come from adding their probabilities?
Your turneasy1 of 3
Write an expression, in terms of n, for the number of outcomes in the sample space when a fair 8-sided die is rolled n times.

Tree Diagrams

  1. How it worksHow do probabilities multiply along one branch and add across separate branches?
  2. Worked exampleFind the chance of drawing two red marbles in a row from 4 red and 6 blue with no replacing.
  3. Common mistakeWhy doesn't the second layer of a tree keep the first layer's numbers when nothing is put back?
  4. Going furtherWhat happens to a tree diagram when a third stage gets added to the process?
Your turnmedium1 of 3
A bag holds 12 red marbles and 6 blue ones. Two marbles are drawn one after the other without replacing the first. Find the probability that both are red.

Independent and Dependent Events

  1. How it worksHow does putting the first item back or leaving it out change every later probability?
  2. Common mistakeWhy isn't a slot machine ever due for a win after a long run of losses?
  3. Worked exampleFind the chance of two aces in a row from a 52-card deck, with replacing and then without.
  4. Going furtherWhat happens to dependence between draws as the pile you are drawing from grows enormous?
Your turneasy1 of 3
A bag holds 15 red marbles and 10 blue ones. Two marbles are drawn, with the first put back before the second is taken. Find the probability that both are red.

Experimental and Theoretical Probability

  1. How it worksHow does long-run relative frequency creep toward the theoretical probability over many trials?
  2. Worked exampleFind the experimental chance of a 3 when 200 rolls give 41 threes, then weigh it against one sixth.
  3. Going furtherWhat happens to the gap between experiment and theory as trials climb from 50 to 5000?
  4. Common mistakeWhy isn't a run of 7 heads in 10 flips proof that a coin has been weighted?
Your turneasy1 of 3
A number cube is rolled 200 times and gives 48 threes. Find the experimental probability of rolling a three.

Designing a Simulation

  1. How it worksHow do you assign digits to outcomes so random numbers copy a real chance process?
  2. Worked exampleFind a digit assignment for a 40 percent chance, then use it to estimate needing 4 donors.
  3. Going furtherWhat happens to a simulation's estimate when you run 20 trials instead of 2000?
Your turneasy1 of 3
In a simulation, the digits 0 through 3 stand for a donor with the needed blood type, giving a 40 percent chance. Over 145 simulated people, how many are expected to have that type?

Relative Frequency and Two-Way Tables

  1. How it worksHow do row totals, column totals and the grand total each give a different relative frequency?
  2. Worked exampleFind the share of dog owners who also own a cat when 42 own dogs and 18 of those own both.
  3. Common mistakeWhy isn't the share of cat owners with dogs the same as the share of dog owners with cats?
  4. Going furtherWhat happens to a relative frequency when you switch from a row total to the grand total?
Your turneasy1 of 3
A survey finds that 42 people own a dog, and 27 of those also own a cat. What percentage of the dog owners also own a cat?