# Representing data: Graphs and charts

## In a nutshell

Graphs are a clear and visual way to represent data. They help you to recognise trends and irregularities, and you can use statistics such as the mean, median and mode to compare the data.

## Charts

### Bar charts

A bar chart represents each item of data as a bar with its height representing the frequency of the item.

##### Example 1

*Plot a bar chart of the following data of participants running in a race and distance ran:*

**DISTANCE RAN$/\text{km}$** | **FREQUENCY** |

$8$ | $60$ |

$10$ | $80$ |

$15$ | $120$ |

$20$ | $80$ |

### Pie charts

A pie chart is visual display where a circle is divided into sectors that each represent a proportion of the whole.

##### Example 2

*Plot a pie chart of the following data of instruments practised by students in a class.*

#### Instrument practised | #### Frequency |

Piano | $6$ |

Guitar | $2$ |

Saxophone | $2$ |

Flute | $1$ |

Drums | $2$ |

Violin | $3$ |

Trumpet | $2$ |

## Graphs

### Scatter graphs

A scatter graph is a plot of the data where the position of each piece of data is represented by its value on each axis.

##### Example 3

*Plot a scatter graph of the following data:*

*$\begin{array}{c|c:c:c:c:c:c:c:c} x & 1 &2 &3 &5 &7 &9 &11 & 12 \\ \hline y & 10 & 8 & 11 & 7 & 6 & 9 & 12 & 14 \end{array}$*

### Line graphs

A line graph is a scatter plot of the data with lines between each point, where each point represents the data of a unit.

##### Example 4

*Plot a line graph of the following data:*

*$\begin{array}{c|c:c:c:c:c:c:c:c:c} x &0 &1 & 2 & 3 & 5 & 7 & 9 & 11 & 12 \\ \hline y & 60 & 75 & 84 & 92 & 112 & 116 & 134 & 140& 144 \end {array}$*

## Statistics

### Definition

A statistic is any calculation derived from data.

### Common statistics

**Mean** | The arithmetic mean of all the data. |

**Median** | $50\%$ of the data lies below this value, $50\%$ of the data lies above this value. |

**Mode** | The most common value within the data. |

**Range** | The difference between the greatest and smallest values of all the data. |

**Lower quartile** | $25\%$ of the data lies below this value. |

**Upper quartile** | $25\%$ of the data lies above this value. |

**Interquartile range** | The difference between the upper and lower quartiles. |

##### Example 5

*For the following data, work out the mean, median, mode, range and interquartile range. Plot a box-plot diagram of the data.*

*$23 \quad 9 \quad 26 \quad 36 \quad 26 \quad 11 \quad 21 \quad 12 \quad 35 \quad 10 \quad 14 \quad 18$*

*Mean**:* $\dfrac{23+9+26+36+26+11+21+12+35+10+14+18}{12} = \underline{20.1} \thinspace(3\thinspace\text{s.f.})$

*Median**: The data is ordered as follows:*

*$9 \quad 10 \quad 11 \quad 12 \quad 14 \quad 18 \quad 21 \quad 23 \quad 26 \quad 26 \quad 35 \quad 36$*

*The median is the value in the middle of this* $\dfrac{18+21}{2}=\underline{19.5}$

*Mode**: The most common value *$= \underline{26}$

*Range**: *$36 -9 = \underline{27}$

*Lower quartile:** *$\underline{11.5}$

*Upper quartile:** *$\underline{26}$

*Interquartile range:** *$26 - 11.5 = \underline{14.5}$

*A box-plot diagram shows the lowest and greatest values, and the three quartile values as follows:*