Statistics

Types of tables, charts and diagrams

Two-way table

The total at the end of each row and column is the total for that individual row or column.

Tally table

Each mark (|) represents one of that thing. One complete group (shown here with a line through it) represents 5.

Frequency is the number of times a particular value or event occurs within a set of data.

Bar charts

In a bar chart, the bars are separated.

The bar chart below is called a stacked bar chart. This is used to show how a total is split.

The last type of bar graph is the dual (side-by-side) bar graph. It is used when comparing two different groups side by side.

Pie charts

Pie charts show how a whole is divided into parts. Each slice represents a category's share out of the total.

(i) Running is $135^\circ$ out of $360^\circ$, and that is 90 people. Let the total be $n$:

$\frac{135}{360} \times n = 90 \quad\Rightarrow\quad n = 90 \times \frac{360}{135} = 240$

(ii) Angles around a point add up to $360^\circ$. The cycling slice has a right angle ($90^\circ$). So the yoga angle is:

$360 - (135 + 90 + 60) = 360 - 285 = 75^\circ$

$\text{Fraction} = \frac{75}{360} = \frac{5}{24}$

Pictograms

Pictograms use pictures or icons to represent data. There will always be a key that tells what each symbol represents.

Key: 1 symbol $= 2$ fish.

Scatter diagrams

A scatter diagram is a graph that shows the relationship between two variables by plotting points. Each point represents one person or thing and has two values.

Draw a line of best fit: a straight line that goes through the middle of the points, to check the trend of the points.

In this example, the line of best fit shows a negative correlation, which means that an increase in one thing means a decrease in the other.

Histograms

A histogram shows grouped data. The bars touch in this graph because the data is continuous. Continuous data is divided into intervals and then used in the histogram to make better sense of the data.

(i) Use the formula for frequency density. For example, $8 < n \le 10$ has $\frac{16}{2} = 8$. The bar heights are 3, 8, 7, 5 and 1.2.

(ii) We only want values from 11 to 20.

• In $10 < n \le 12$, only part of the interval (from 11 to 12) counts. Half of the frequency in this group belongs to $11 < n \le 12$, so we take half of $14 = 7$ days.
• From 12 to 15: all values count, $15$ days.
• From 15 to 20: all values count, $6$ days.

$\frac{7 + 15 + 6}{60} \times 100 = 46.7\%$

Cumulative frequency diagrams

A cumulative frequency diagram shows how many data values are less than or equal to a certain number. It helps us see totals building up.

For example, this graph shows us that most of the people's heights lie between 1.3 and 1.4 m (the curve is steepest there).

This graph is useful in determining:

• Median: halfway up the $y$-axis.
• Lower quartile ($Q_1$): $\frac14$ of the total frequency.
• Upper quartile ($Q_3$): $\frac34$ of the total frequency.
• Inter-quartile range $= Q_3 - Q_1$.

(a) Median. $\frac{80}{2} = 40$th value. Read across from 40 on the vertical axis to the curve, then down: 1.33 m.

(b) 30th percentile. $\frac{30}{100} \times 80 = 24$th value. Read across from 24 to the curve, then down: 1.28 m.

(c) Value of $y$. $\frac25 \times 80 = 32$ trees are in Class I, so the other $80 - 32 = 48$ trees are not. Read across from 48 to the curve, then down: $y =$ 1.36 m.

(d) Frequency table. The frequency is the difference between consecutive cumulative frequencies, read from the curve at $h = 1.1, 1.2, 1.3, 1.4, 1.5, 1.6$:

(For example, $58 - 30 = 28$ and $80 - 76 = 4$.)

Because there are 2 unknown variables, we need to make simultaneous equations.

The frequencies add up to 50: $p + 15 + 17 + q = 50$, so

$p + q = 18 \qquad \text{(1)}$

The mean is the sum of $\text{frequency} \times \text{midpoint}$ divided by the total:

$1.81 = \frac{1.6p + 1.75(15) + 1.85(17) + 2.1q}{50}$

$1.81 \times 50 = 1.6p + 26.25 + 31.45 + 2.1q \quad\Rightarrow\quad 1.6p + 2.1q = 32.8 \qquad \text{(2)}$

Solving: from (1), $p = 18 - q$. Substituting into (2): $1.6(18 - q) + 2.1q = 32.8$, so $28.8 + 0.5q = 32.8$, so $q = 8$ and $p = 10$.

Frequency polygons

A frequency polygon is a line graph made by joining the midpoints of the top of each bar from a bar chart or histogram. It is used to show trends or patterns in data and is great for comparing the shapes of data.

Averages and spread

Mean

The mean is one of the ways to get the average of a dataset. You add all the numbers and divide by how many there are.

$\text{Mean} = \frac{140 + 145 + 150 + 150 + 155}{5} = \frac{740}{5} = 148\text{ cm}$

Median

The median is the middle number when the data is in order. Use the formula below to get the position of the median term.

If the answer has $.5$ in it, take the average of the two terms either side. For example, if the position is 4.5, take the average of the 4th and 5th terms.

The position is $\frac12(5 + 1) = 3$, so the median is the 3rd value: 150 cm. (Cut one term from each side repeatedly.)

Mode

The value that comes up the most.

150 appears twice, so the mode is 150 cm.

Range

It is the difference between the biggest and smallest value. It shows how spread out the data is.

$\text{Range} = 155 - 140 = 15\text{ cm}$

Mean of grouped data

When values are in groups, we don't know the exact numbers. So we use the midpoint of each group.

$\sum fx = 535, \quad \sum f = 25 \quad\Rightarrow\quad \text{Mean} = \frac{535}{25} = 21.4$

Modal class

The class (range) with the highest frequency. From the table above, the highest frequency is 10, so the modal class is 20–30.

Cumulative frequency table

We add up frequencies step by step to get a cumulative frequency table.

Quartiles and IQR

• $Q_1$ (lower quartile): the 25% point, term $= \frac14(n + 1)$.
• $Q_2$ (median): the 50% point.
• $Q_3$ (upper quartile): the 75% point, term $= \frac34(n + 1)$.
• Inter-quartile range $= Q_3 - Q_1$. It shows the spread of the middle 50% of the data.