Data, Statistics, and Probability

🔑 Key ideas

• Data are facts and numbers. Tables and graphs organise them so you can use them.
• Always read the title and every label first, so you know what the numbers mean.
• If a graph uses symbols, a key tells you what each symbol is worth.

📘 Example 1 · Which county had the greatest percent increase in population?

1. Read the labels. The question is about population, so ignore the land-area column.
2. Look at the data. Only Adams and Davis grew from 2000 to 2010. The others shrank.
3. Use the data. Percent increase = increase ÷ starting amount. Adams: 4,167 ÷ 11,128 ≈ 0.37, about 37%. Davis: 8,630 ÷ 82,204 ≈ 0.10, about 10%.

Answer: Adams County. Davis gained more people, but Adams grew by a bigger percent.

📘 Example 2 · How many books did the South and West branches check out together?

1. South has 4½ symbols and West has 6. Together: 4½ + 6 = 10½ symbols.
2. Each symbol is 150 books: 10.5 × 150 = 1,575 books.

🔑 Key ideas

• A bar graph uses bars to show amounts. Compare the end of each bar with the number scale.
• A double-bar graph compares two sets of data. The key tells you which bar is which.
• A line graph shows change over time. A steeper line means a faster change.

📘 Example 3 · How many more repairs were there in February than in January?

The January bar reaches 20. The February bar stops a little above the 30 line, at 32. 32 − 20 = 12 more repairs.

📘 Example 4 · Film B's cost is what percent of its ticket sales?

1. Read the key. Solid bars are cost; outlined bars are ticket sales. The scale is in millions of dollars.
2. Read the bars. Film B cost $30 million and took $65 million.
3. Use the data. 30 ÷ 65 ≈ 0.46, so the cost was about 46% of ticket sales.

📘 Example 5 · Between which two months did visitors increase the most?

The steepest upward piece of line is from June to July: 260 − 180 = 80 more visitors. (May to June was only 60 more.)

🔑 Key ideas

• The whole circle is the whole amount (100%). Each section is a part of it.
• Sections are usually labelled with percents, but they can show fractions, decimals or amounts.
• The size of a section matches its share: a 25% section is 14 of the circle.

📘 Example 6 · How much of the budget goes on rent?

1. Read the title. The whole circle is the $2,000 budget.
2. Find the section. Rent is 40%.
3. Use the data. 40% of $2,000 = 0.40 × 2,000 = $800.

🔑 Key ideas

• A frequency table shows how often each item appears, often with tally marks.
• Numbers can be grouped into intervals such as 18–24 and 25–40.
• The mean, median and mode each describe the “middle” of a data set. The range describes its spread.

📘 Example 1 · What is the ratio of black cars to silver cars?

Black has 27 marks and silver has 9. Write the ratio and reduce: 27 : 9 = 3 : 1. Black was chosen 3 times as often as silver.

📘 Example 2 · What percent of the customers were 25 to 40 years old?

1. Total customers: 6 + 14 + 28 + 12 + 10 = 70. That is the whole.
2. The part is 28. 28 ÷ 70 = 0.4 = 40%.

📘 Example 3 · The mean of 14, 21, 3, 20 and 10 points

Add: 14 + 21 + 3 + 20 + 10 = 68. There are 5 games, so 68 ÷ 5 = 13.6 points per game.

No team scores 13.6 points in a game, but the mean still tells you the centre of the scores.

📘 Example 4 · The median of 43, 28, 24, 36, 32, 37, 48 sales

In order: 24, 28, 32, 36, 37, 43, 48. There are 7 values, so the 4th is in the middle. The median is 36.

📘 Example 5 · The median of 90, 72, 88, 94, 91, 80 (an even count)

In order: 72, 80, 88, 90, 91, 94. The two middle values are 88 and 90. (88 + 90) ÷ 2 = 89.

📘 Example 6 · The mode and range of 45°, 44°, 45°, 47°, 46°, 45°

45° appears 3 times, more than any other, so the mode is 45°. The range is 47° − 44° = 3°.

🔑 Key ideas

• A line plot shows how often each value appears, with an X or a dot above a number line for every item.
• It is a frequency table drawn as a picture, so you can see the distribution (how the data are spread out).
• Look for the highest frequency (the tallest stack), the range and any outliers.

🔑 Key ideas

• A histogram shows counts for groups (intervals) of numbers, such as weeks or score ranges.
• The groups go along the horizontal axis and the frequencies up the vertical axis.
• The bars touch, because the groups follow on from each other with no gaps.

📘 Example 1 · What percent of the appointments were in the first two weeks of March?

1. Total appointments: 2 + 4 + 8 + 11 + 14 + 17 + 13 + 7 + 5 = 81.
2. First two weeks: 2 + 4 = 6. 6 ÷ 81 ≈ 0.074, about 7%.

🔑 Key ideas

• Probability is a ratio. You can write it as a ratio, a fraction, a decimal or a percent.
• Theoretical probability = favourable outcomes ÷ possible outcomes.
• Experimental probability = times it happened ÷ number of trials.

📘 Example 1 · A spinner has 8 equal sections, and 2 are labelled 4. What is P(4)?

2 favourable out of 8 possible: 28 = 14 = 0.25 = 25%, or 1 out of 4.

📘 Example 2 · Experimental probability

Two players spun the same spinner 20 times and got: 2, 4, 4, 6, 4, 3, 4, 6, 4, 3, 1, 6, 2, 2, 5, 2, 4, 2, 1, 2.

A 4 came up 6 times in 20 trials: 620 = 310 = 0.3 = 30%.

That is close to the theoretical 25%, but not equal. Theory tells you what will probably happen, not what will actually happen.

📘 Example 3 · Two coins. What is P(both heads)?

The outcomes are HH, HT, TH, TT, and only HH is favourable: 1 out of 4. By multiplying: 12 × 12 = 14 = 25%.

📘 Example 4 · A box has 4 blue and 2 red marbles. You take two. What is P(both blue)?

1. First marble: 4 blue out of 6, so 46 = 23.
2. If it was blue, 5 marbles are left and 3 are blue: 35.
3. Multiply: 23 × 35 = 615 = 25, or 2 out of 5.

If the first marble were put back before the second draw, the events would be independent: 23 × 23 = 49.

🔑 Key ideas

• A combination is a group chosen from a larger set when the order doesn't matter.
• Choosing several items from one group: use an organised list or a table.
• Choosing one item from each of several groups: use a tree diagram or the fundamental counting principle.

📘 Example 1 · Pablo buys 2 different fruits from apples, bananas, grapes and pears

Apples-and-bananas is the same pair as bananas-and-apples, so order doesn't matter. List them in order so you don't repeat any:

AB  AG  AP      (pairs that start with apples)
    BG  BP      (start with bananas, skip BA)
        GP      (start with grapes)

There are 6 combinations. Nothing new starts with P, because every pair with P is already listed.

📘 Example 2 · Sarah has 3 shirts, 2 pairs of pants and 2 pairs of shoes. How many outfits?

A tree diagram would show each shirt branching into 2 pants, and each of those into 2 shoes. Counting the end branches gives 12.

The fundamental counting principle is faster: multiply the number of choices in each group. 3 × 2 × 2 = 12 outfits.

📘 Example 3 · 3 starters, 6 main courses and 4 desserts. How many three-course meals?

3 × 6 × 4 = 72 meals.

🔑 Key ideas

• A permutation is an arrangement where the order matters.
• Multiply the number of choices for the first place, then the second, and so on, one fewer each time.
• Stop multiplying when every place is filled.

📘 Example 4 · Eliza has 4 things to do on her day off. In how many orders can she do them?

4 choices for first, then 3, then 2, then 1: 4 × 3 × 2 × 1 = 24 orders.

📘 Example 5 · 10 runners. How many ways can 1st, 2nd and 3rd place be filled?

10 can win first, then 9 are left for second, then 8 for third. Three places, so stop after three numbers: 10 × 9 × 8 = 720. The test may write this as P(10, 3).

Note that 10 × 3 = 30 is wrong, and so is 10 × 9 × … × 1, which would order all ten runners.

🔑 Key ideas

• The same proportion can be shown as a number, in a table, on a graph or as a ratio. Its size doesn't change.
• Two proportions are easiest to compare when they are in the same form.
• Usually the quickest way is to turn each one into a number: a percent, a decimal or a rate.

📘 Example 1 · Which class has the greater share of walkers? Class C says “3 in every 8 of us walk.”

1. Class A (table): 10 ÷ 25 = 0.40 = 40%.
2. Class B (graph): read the Walk bar as 9, then 9 ÷ 30 = 0.30 = 30%.
3. Class C (ratio): 3 ÷ 8 = 0.375 = 37.5%.

Answer: Class A has the greatest share (40%), then Class C, then Class B. Class B has fewer walkers than A only by 1 student, but its share is 10 percentage points lower because the class is bigger.

📘 Example 2 · Comparing rates

A new car goes from 0 to 60 miles per hour (mph) in 8 seconds. A graph shows an older car reaching 60 mph in 6 seconds, and a table shows a third car gaining 7 mph every second.

1. Write every car as a rate in mph per second. New car: 60 ÷ 8 = 7.5 mph/s.
2. Older car: 60 ÷ 6 = 10 mph/s. Third car: 7 mph/s.
3. Compare: the new car speeds up faster than the third car, but more slowly than the older one.