Math 7 · Data and probability
The middle of the data, and the chance of two things
One number can stand for a whole set of data, but the mean, median and mode each choose it differently, and one unusual value can pull them apart. A circle graph shows how a whole splits into parts. Probability measures how likely something is — first by counting what could happen, then by trying it.
- 1. The words, first
- 2. Mean, median, mode and range
- 3. Circle graphs
- 4. Probability of two events
- 5. What costs marks
The words, first
The idea: Outcome and sample space are the probability words to be sure of: one thing that can happen, and the list of all of them.
| Word | What it means |
|---|---|
| Mean | Add the values and divide by how many there are. |
| Median | The middle value once the data is in order. With an even number of values, halfway between the middle two. |
| Mode | The value that appears most often. A set can have one mode, several, or none. |
| Range | The largest value minus the smallest. It measures spread, not the middle. |
| Outlier | A value far away from the rest of the data. |
| Circle graph | A circle cut into sectors (slices), each showing a part of one whole as a percent. |
| Probability | How likely something is, from 0 (impossible) to 1 (certain), as a fraction, decimal, percent or ratio. |
| Outcome, sample space | One possible result, like rolling a 4; and the list of every possible outcome. A favourable outcome is one you are counting. |
| Independent events | Events where one does not change the chances of the other, like rolling two dice. |
| Theoretical, experimental | Worked out by counting equally likely outcomes; or found by doing it: how often it happened ÷ how many tries. |
Mean, median, mode and range
The idea: The mean shares everything out equally, the median is the middle, and the mode is the most common. The range is not a middle at all: it measures spread.
Worked example. Seven students count the books they read in a month: 2, 3, 3, 4, 5, 6, 26.
- Mean: 2 + 3 + 3 + 4 + 5 + 6 + 26 = 49, and 49 ÷ 7 = 7.
- Median: the values are already in order, and the middle (4th) one is 4.
- Mode: 3 appears twice, more than any other value: 3.
- Range: 26 − 2 = 24.
What the outlier did. Without the 26, the mean would be 23 ÷ 6, about 3.8, and the median 3.5. One student pushed the mean up to 7 while the median barely moved. But six of the seven read 6 books or fewer, so the median, 4, is the fairer summary.
An even number of values. For 3, 5, 8, 10 the median is halfway between 5 and 8: (5 + 8) ÷ 2 = 6.5.
Which to use. The mean when the values are fairly even. The median when there is an outlier, as with house prices. The mode for the most common value: a shoe store orders the most common size.
Circle graphs
The idea: Each sector shows a percent of the whole, and its central angle is that percent of 360°.
Worked example. 40 students were asked how they get to school.
| Way | Students | Percent | Central angle |
|---|---|---|---|
| Bus | 14 | 14 ÷ 40 = 0.35 = 35% | 0.35 × 360° = 126° |
| Car | 12 | 12 ÷ 40 = 0.3 = 30% | 0.3 × 360° = 108° |
| Walk | 10 | 10 ÷ 40 = 0.25 = 25% | 0.25 × 360° = 90° |
| Bike | 4 | 4 ÷ 40 = 0.1 = 10% | 0.1 × 360° = 36° |
| Total | 40 | 100% | 360° |
Two checks before you draw. The percents must add to 100% and the angles to 360°. And 25% is always a right angle.
Reading one backwards. In a circle graph of 200 people, a 90° sector is 90 ÷ 360 = 0.25 of the circle: 25%, or 50 people.
Probability of two events
The idea: List every outcome in a table or a tree, count the ones you want, and divide by the total. With independent events, one result does not change the chances of the other.
probability = favourable outcomes ÷ possible outcomes, when every outcome is equally likely
Worked example. Two dice are rolled and the numbers added. What is the probability of a sum of 7?
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
- Sample space: 6 rows × 6 columns = 36 equally likely outcomes.
- Count the 7s: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1. That is 6.
- Probability: 6/36 = 1/6, about 0.167 or 16.7%. As a ratio of favourable to possible outcomes, 1 : 6.
The trap. There are 11 possible sums, from 2 to 12, but they are not equally likely: one outcome gives 2, and six give 7. Saying the chance of 7 is 1/11 counts sums, not outcomes.
A tree works too. Flip a coin, then spin a spinner with three equal parts: red, blue and green. Two branches each split into three, making 6 outcomes, so the chance of heads and red is 1/6.
Theoretical and experimental. Roll two dice 50 times and get a 7 eleven times: the experimental probability is 11/50 = 22%, not 16.7%. Fifty rolls is not many; over hundreds, the result usually settles near 1/6. And dice have no memory: after three 7s in a row, the chance of another is still 1/6.
What costs marks
The idea: Two are about the middle of the data, one is about circle graphs, and two are about probability.
- Calling a mean pulled up by an outlier “typical”. Use the median.
- Finding the median before putting the data in order.
- Using the percent as the angle. 35% is 126°, not 35°.
- Treating the 11 sums of two dice as equally likely. Count the 36 outcomes.
- Expecting an experiment to match the theory exactly. It gets closer with more tries.