# Graphical Representations

Graphical representations of data allow a visual aid in reading and interpreting data.

## Stem-and-Leaf Plot

A stem-and-leaf plot is a method of displaying data by separating the last digit of each numerical element to create a stem and leaf shape.

**Example:**

The following data is converted to the following stem and leaf plot:

**Data**- 2, 4, 10, 10, 12, 14, 15, 15, 15, 23, 24, 24, 27, 28, 34, 37, 39, 39, 47
**Plot**- 0 | 2 4
- 1 | 0 0 2 4 5 5 5
- 2 | 3 4 4 7 8
- 3 | 4 7 9 9
- 4 | 7

A stem-and-leaf plot allows the following understanding of the data:

- finding typical values
- the spread of the values
- gaps in the data
- the symmetry of the distribution values
- number and location of peaks
- outliers

Stem-and-leaf plots can be modified to make the data easier to read or compare with other data.

- splitting the stem, as in 0 - 4 and 5 - 9 are split into two different stems, to increase the detail in the distribution
- back-to-back or comparative stem-and-leaf plots, where similar data sets are shown on either side of a stem for easy comparison

**Example:**

This is a comparative stem-and-leaf plot, with separate sets of data on each side.

The plot shows that the data set on the right has a greater amount of large values than the data on the left.

## Bar Charts and Histograms

Bar charts and histograms count the frequency of the data in certain categories and plots these as rectangular bars. These categories can be of a categorical variable, such as a frequency of people with black, brown, blonde or white hair for example. This would require a **bar chart**.

The categories can be discrete numerical variables, such as the frequency of the amount of cars households own. This would require a **histogram**.

The categories can be numerically continuous, requiring categories to represent a range of values, such as the range of temperatures over a day. This also requires a histogram, though it should be noted that the bars now touch, unlike the discrete categories. To work out the size of these categories, known as classes, divide the range of data by the square root of the number of observations.

**Example:**

- Smallest data value = 20
- Largest data value = 150
- Range = 150 - 20 = 130
- Number of observations = 55
- Number of classes = sqrt(55) = 7.42 = 7
- Size of classes = 130/7 = 18.57 = 19

The shape of the histogram can show patterns in the data. These include:

**Symmetry:** The frequency distribution is symmetrical from the centre.

**Skewed:** The frequency has the majority of frequency on either the left or right.

**Left skewed:**The majority of the frequency is on the right, with data tailing off to the left.

**Unimodal/Bimodal/Multimodal:** A number of peaks in the frequency of the data.

**Bell-shaped:** The frequency distribution is both symmetric and unimodal.

The size of the classes can be unequal. Combining classes can be done, though it gives a distorted representation of the data. Therefore a **density histogram** is used. This is where the frequency of the class is divided by the total number of observations to give the y-axis value.

## End

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