International Baccalaureate · PYP 5

Statistics: Graphs and Averages

⌚ ~202 min ▢ 23 blocks ✓ 93 exercises ➣ Prerequisites : Percentages and Ratios, Probability, Operations with Decimals, Data Handling, Data Handling, Data Handling

I Population and Types of Data

Suppose a class wants to investigate the question: “What is our favourite school subject?” The survey covers all 25 students in the class.

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Definition — Population

A population is the entire group of people or things that you want to study.

Example

For our survey, the population is all 25 students in our class.

Example

The data we collect will be the list of answers from each student, such as “Maths”, “Art”, “Science”, “Art”, “Sport”, etc.

Definition — Qualitative and Quantitative Variables: Counting or Measuring

A survey looks at a variable: what is observed for each person or each object.

a qualitative variable (the answers are categorical data): the answer names a category (eye colour, favourite food), even when it is written with a number, like a shirt number;
a quantitative variable (the answers are numerical data): the answer says how many. It is counted (the number of siblings) or measured (height in centimetres, mass in kilograms, a duration).

To decide, ask: does the answer say how many?

Example

For our survey:

The variable “Favourite Subject” is qualitative because the answers are categories (Maths, Art, etc.).
The variable “time spent on homework each week” is quantitative: the answer is a duration, which is measured (e.g., 2 hours, 5 hours).

Skills to practice

Identifying Populations
Classifying Data Types
Designing Questions for Data Types

II Frequency Tables

Definition — Frequency and Relative Frequency

The frequency (or count) of a category is how many times that category appears in our data.
Relative frequency is the proportion of the data that falls into a category. We can write it as a fraction, a decimal, or a percentage. $$ \text{Relative frequency (in }\pourcent\text{)} = \frac{\text{Frequency}}{\text{Total number of data values}} \times 100\pourcent $$

Method — Creating a Frequency Table

To organise our data, we use a frequency table. We use tally marks to count the frequency for each category. Then, we can calculate the relative frequency.
Here is the data for the “Favourite Subject” survey of 25 students, organised into a frequency table:

Subject	Tally Marks	Frequency (Count)	Relative Frequency
Maths		8	$8/25 = 32\pourcent$
Science		5	$5/25 = 20\pourcent$
Sport		7	$7/25 = 28\pourcent$
Art		5	$5/25 = 20\pourcent$
Total		25	100$\pourcent$

From this table, we can easily see which subjects are more or less popular. For example, Maths has the highest frequency (8 students), so it is the most popular subject.

Example

Here is a bar chart showing the data from our “Favourite Subject” table: the number of students for each subject (also called its frequency). It helps us see the results at a glance.

Subject	Number of Students
Maths	8
Science	5
Sport	7
Art	5

From this chart, we see at once that Maths is the most popular subject and that Science and Art are equally popular.

Skills to practice

Finding Frequencies in Lists and Bar Charts
Finding Relative Frequencies

III Data in a Text

Method — Taking data out of a text

A problem does not always come with a table: the data can be hidden in a text. Before answering anything, take the data out of the text and put it in a table.

Read the text once, then underline every number together with what it counts (“40 comics”, “8 picture books”). A number without its category is useless.
Draw a table with one row per category, and a row for the total.
Fill it in. Be careful: some numbers in a text are not the data we are looking for (a date, a page number, an age), and some data is given indirectly (“the others”, “twice as many”): compute it. Sometimes, on the contrary, a piece of data we need is not in the text at all: then we cannot answer without it.
Check the total, then answer the question from the table, not from the text.

Example

“In March, the school library lent out 96 books. The pupils borrowed 40 comics, 32 novels, some non-fiction books and 8 picture books. The library, which opened in 2009, is open 3 days a week.” How many non-fiction books were borrowed? Which kind of book was borrowed twice as often as the picture books?

Answer

The numbers that count books are $96$ (total), $40$, $32$ and $8$; the year $2009$ and the $3$ days are not the data we need. The non-fiction books are given indirectly: $96 - 40 - 32 - 8 = 16$.

Kind of book	Number
Comics	40
Novels	32
Non-fiction	$96-40-32-8 = 16$
Picture books	8
Total	96

Twice $8$ is $16$: the non-fiction books were borrowed twice as often as the picture books.

Skills to practice

Finding the Missing Information
Taking Data from a Text into a Table

IV The Pie Chart

Definition — Pie chart

A pie chart shows how a total is shared out between categories. The disc stands for the whole total; each category gets a sector (a slice), and the bigger the sector, the bigger the share. Half of the disc means half of the total, a quarter of the disc a quarter of the total. On our charts, the disc is divided into $12$ equal parts by thin lines, so a sector can be read as a number of parts out of $12$: $3$ parts out of $12$ is a quarter, $2$ parts is a sixth (because $6 \times 2 = 12$), $6$ parts is a half. A semi-circular chart works the same way, with a half-disc that stands for the whole total; ours are divided into $10$ equal parts.

Method — Reading a pie chart

To find how many things a sector stands for:

Find the total. It is written in the title or in the text (“the favourite sport of the $24$ pupils”). Without the total, a pie chart like ours, with nothing written in the sectors, only gives shares, not numbers.
Read the share of the sector: count its parts out of $12$, or compare it with a half or a quarter of the disc.
Compute the number: the share of the total. A quarter of $24$ is $24 \div 4 = 6$; one part out of $12$ is $24 \div 12 = 2$, so $2$ parts are $4$.
Check: the numbers of all the sectors must add up to the total.

Example

The pie chart shows the favourite sport of the $24$ pupils of a class. How many pupils prefer each sport?

Answer

The total is $24$.

Football: 6 parts out of 12, that is half of the disc, so half of 24 $= 12$ pupils.
Swimming: 3 parts out of 12, that is a quarter of the disc, so a quarter of 24 $= 6$ pupils.
Dance: 2 parts out of 12, that is $\frac{1}{6}$ of the disc, so $\frac{1}{6}$ of 24 $= 4$ pupils.
Judo: 1 part out of 12, that is $\frac{1}{12}$ of the disc, so $\frac{1}{12}$ of 24 $= 2$ pupils.

Check: $12 + 6 + 4 + 2 = 24$.

Example

This semi-circular chart shows how the $40$ pupils of two classes come to school. The half-disc is divided into $10$ equal parts. How many pupils come by bike? On foot?

Answer

The half-disc stands for the $40$ pupils, and one part out of $10$ is $40 \div 10 = 4$ pupils.

On foot: 5 parts out of 10 of the half-disc, that is half of the total, so 20.
Bike: 3 parts out of 10 of the half-disc, that is $\frac{3}{10}$ of the total, so 12.
Bus: 2 parts out of 10 of the half-disc, that is $\frac{1}{5}$ of the total, so 8.

Check: $20 + 12 + 8 = 40$.

Shares, not numbers. These two pie charts look exactly the same: in both, reading takes a quarter of the disc. But the first one is about a class of $20$ pupils and the second about a school of $100$ pupils. A quarter of $20$ is $5$ pupils; a quarter of $100$ is $25$ pupils. The same sector can stand for very different numbers: shares can be compared on the discs, but to compare numbers between two pie charts, look at their totals first.

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Skills to practice

Judging Statements about Pie Charts
Reading a Pie Chart
Turning a Pie Chart into a Table

V The Mean

Definition — The Mean

Four children build towers of cubes: $6$, $2$, $5$ and $3$ cubes. They move cubes from the tall towers to the short ones until all the towers are the same height. No cube is added or taken away, so the $16$ cubes are shared out equally between the $4$ towers: each tower ends with $16 \div 4 = 4$ cubes.

The mean of a set of numbers is the value that each number would have if the total were shared out equally between them: $$\text{mean} = \text{total of the values} \div \text{number of values}.$$ The mean is found only for numbers, never for words such as colours or pets. It is often called the average.

Method — Finding the Mean

Add all the values: this gives the total. A value $0$ is a value too: it adds nothing to the total, but it counts as one value.
Count the values.
Divide the total by the number of values. The result is the mean.
Check: the mean lies between the smallest and the largest value. It is not always one of the values, and not always a whole number.

The other way round, when the mean and the number of values are known: $\text{total} = \text{mean} \times \text{number of values}$.

Example

In five matches, a team scored $3$, $1$, $4$, $0$ and $2$ goals. Find the mean number of goals per match.

Answer

The $0$ is a value: there are $5$ matches, not $4$. $$\begin{aligned}[t] \text{mean} &= (3 + 1 + 4 + 0 + 2) \div 5 && \text{(total} \div \text{number of values)}\\ &= 10 \div 5\\ &= 2. \end{aligned}$$ The mean is 2 goals per match: with $2$ goals in each of the $5$ matches, the team would have scored the same $10$ goals. Check: $2$ lies between $0$ and $4$.

Example

Leo scored $12$, $9$, $15$ and $10$ points in four basketball games. Find his mean score per game.

Answer

$$\begin{aligned}[t] \text{mean} &= (12 + 9 + 15 + 10) \div 4 && \text{(total} \div \text{number of values)}\\ &= 46 \div 4\\ &= 11.5. \end{aligned}$$ Leo's mean score is 11.5 points per game. The mean is not one of the four scores, and it is not a whole number: no game ended with $11.5$ points, but $4$ games of $11.5$ points would make the same total, $46$. Check: $11.5$ lies between $9$ and $15$.

Skills to practice

Understanding the Mean
Calculating the Mean
Solving Problems with the Mean

VI The Median

Definition — The Median

Five children, all of different heights, stand in a line from the shortest to the tallest. The child in the middle has two children shorter and two children taller: this child's height is the middle height.
To find the median of a set of numbers, first write the values in order, from the smallest to the largest. The median is the middle of the ordered values: it splits the values into two groups of the same size, one on each side.

With an odd number of values ($3$, $5$, $7$, …), the median is the value in the middle.
With an even number of values ($2$, $4$, $6$, …), two values share the middle: the median is halfway between them. Add the two middle values and divide by $2$.

Method — Finding the Median

Write the values in order, from the smallest to the largest, and count them: the ordered list must have as many values as the data.
Cross off the smallest and the largest value, then again the smallest and the largest of those left, until one or two values are left.
One value left: it is the median. Two values left: the median is halfway between them, their total divided by $2$.

Example

Seven children counted the books they read in one month: $4$, $9$, $2$, $7$, $4$, $12$ and $5$. Find the median.

Answer

In order: $2,\ 4,\ 4,\ 5,\ 7,\ 9,\ 12$ — seven values, as in the data. Seven is odd, so one value is in the middle: crossing off $2$ and $12$, then $4$ and $9$, then $4$ and $7$ leaves $5$. $$2,\ 4,\ 4,\ \underline{5},\ 7,\ 9,\ 12$$ The median is 5 books: three children read fewer books, three read more. Taken in the order of the data, the middle value would be $7$: that is why the values are put in order first.

Example

Six bean plants measure $14$, $9$, $11$, $20$, $8$ and $12$ centimetres. Find the median height.

Answer

In order: $8,\ 9,\ 11,\ 12,\ 14,\ 20$ — six values. Six is even, so two values share the middle: $11$ and $12$. $$\begin{aligned}[t] \text{median} &= (11 + 12) \div 2 && \text{(halfway between the two middle values)}\\ &= 23 \div 2\\ &= 11.5. \end{aligned}$$ The median height is 11.5 cm: three plants are shorter, three are taller.

Skills to practice

Recognising the Median
Finding the Median

VII The Range

Definition — The Range

The range of a set of numbers is the largest value minus the smallest value: $$\text{range} = \text{largest value} - \text{smallest value}.$$ The range measures the spread of the data: a small range means that the values are close together, a large range that they are spread out. The range is the distance between the smallest and the largest value: it does not tell what a typical value is.

Example

Mia and Sam each threw a ball five times. Their throws, in metres:

Mia: $12$, $15$, $14$, $13$, $16$;
Sam: $6$, $20$, $14$, $9$, $21$.

Find the mean and the range of each child's throws. Who throws more regularly?

Answer

Mia: the total is $12 + 15 + 14 + 13 + 16 = 70$, so the mean is $70 \div 5 = 14$ m; the range is $16 - 12 = 4$ m.
Sam: the total is $6 + 20 + 14 + 9 + 21 = 70$, so the mean is $70 \div 5 = 14$ m; the range is $21 - 6 = 15$ m.

The two children have the same mean, $14$ m, but Mia's throws are close together (range $4$ m) and Sam's are spread out (range $15$ m): Mia throws more regularly. The mean alone could not show this difference; the range does.

Mode, Mean, Median and Range

The mode, the mean, the median and the range are four measures: each one sums up a whole set of data in a short answer.

The mode is the value that occurs most often (there can be more than one mode, or none). It also works for words, even words that cannot be put in order: the favourite pet, the size a shop sells most.
The mean shares the total out equally. It uses every value, so one value far from the others pulls it towards that value.
The median is the middle of the ordered data: it only looks at the values in the middle. With three values or more, making the largest value much larger does not change it.
The range does not give a typical value: it tells how spread out the values are.

Five children get $\dollar 2$, $\dollar 3$, $\dollar 3$, $\dollar 4$ and $\dollar 18$ of pocket money a week. The mode is $\dollar 3$, the median is $\dollar 3$, the mean is $\dollar 30 \div 5 = \dollar 6$ and the range is $\dollar 18 - \dollar 2 = \dollar 16$. Four of the five children get less than the mean: the $\dollar 18$ pulls the mean up. Here the median, $\dollar 3$, describes what a typical child gets better than the mean.

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Skills to practice

Comparing Spread and Choosing a Measure
Finding the Range
Choosing and Comparing Measures

VIII The Spreadsheet

Definition — Spreadsheet and Cells

A spreadsheet is a computer program that shows a grid of boxes called cells. The columns are named by letters (A, B, C, …) and the rows by numbers (1, 2, 3, …). A cell is named by the letter of its column followed by the number of its row: the cell B3 is in column B and in row 3. A cell can hold a word, a number or a formula.

Method — Writing a Formula

Type the data, one value in each cell: the names (the days, the pets…) in one column and their numbers in the next column, with a title at the top of each column, in row 1.
Click the cell where the result must appear.
Type a formula. A formula always starts with the sign =:
- =B2+B3 adds the numbers of the cells B2 and B3;
- =SUM(B2:B6) adds all the numbers from B2 to B6 (B2:B6 means the five cells B2, B3, B4, B5 and B6);
- =AVERAGE(B2:B6) gives the mean of these numbers.
Press Enter: the cell shows the result. When the cell is selected, its formula can be read in the formula bar, above the grid.
When a number used by a formula changes, the spreadsheet computes the result again by itself.
To draw a bar chart, select the names and their numbers with the two titles, and nothing else (no total, no mean), then choose a bar chart: each name gets a bar, and each number gives the height of its bar.

Example

Every day of a school week, a class writes down in a spreadsheet how many minutes it spent reading. The teacher types =SUM(B2:B6) in the cell B7 and =AVERAGE(B2:B6) in the cell B8. The grid below is not what the screen shows: it shows what the teacher typed in each cell. On the screen, once Enter is pressed, each of these two cells shows its result instead.

What do the cells B7 and B8 show after pressing Enter?
The class made a mistake: on Tuesday it read for $45$ minutes, not $35$. The number in B3 is changed to $45$. What do B7 and B8 show now?
How can the class draw a bar chart of its reading time?

Answer

B7 adds the five cells: $20 + 35 + 25 + 40 + 30 = 150$. B8 shows their mean: $150 \div 5 = 30$. The cells show $150$ and $30$; the formula =SUM(B2:B6) can still be read in the formula bar when B7 is selected.
Nothing has to be typed again: the spreadsheet computes again by itself. The new total is $150 - 35 + 45 = 160$, so B7 shows $160$ and B8 shows $160 \div 5 = 32$.
Select the cells from A1 to B6 (the days and the minutes, with their titles) and choose a bar chart. Each day gets a bar whose height is its number of minutes; if a number changes, its bar changes too. With $45$ minutes on Tuesday:

Skills to practice

Reading a Spreadsheet
Predicting What a Formula Shows
Writing and Checking Formulas

IX Going further

Skills to practice

Conducting a Full Investigation
Solving Two-Step Problems with Charts
Quiz