Descriptive Statistics

Descriptive Statistics

Descriptive statistics are methods for summarising and presenting data in a meaningful way. They provide a way to describe the main features of a dataset, allowing researchers to understand and communicate the data effectively. In this section, you will learn about measures of central tendency, measures of variability, and how to represent data visually.

Measures of Central Tendency

Measures of central tendency are values that represent the centre of a dataset. The three most common measures are the mean, median, and mode.

Mean

The mean is the average of a set of numbers. To calculate the mean, follow these steps:

1. Add all the values together.
2. Divide the total by the number of values.

For example, consider the following dataset: 5, 10, 15, 20, 25.

Step 1: Add the values:

5 + 10 + 15 + 20 + 25 = 75

Step 2: Divide by the number of values (which is 5):

75 ÷ 5 = 15

Thus, the mean of the dataset is 15.

Median

The median is the middle value in a dataset when the numbers are arranged in ascending order. If there is an even number of values, the median is the average of the two middle numbers.

Using the same dataset: 5, 10, 15, 20, 25, we first arrange it in order (it is already in order). The middle value is 15, so the median is:

Median = 15

If we had the dataset: 5, 10, 15, 20, we would find the median as follows:

1. Arrange the values: 5, 10, 15, 20.
2. Since there are 4 values, take the average of the two middle numbers (10 and 15):

(10 + 15) ÷ 2 = 12.5

Thus, the median is 12.5.

Mode

The mode is the value that appears most frequently in a dataset. A dataset may have one mode, more than one mode, or no mode at all. For example, consider the dataset: 5, 10, 10, 15, 20.

In this case, the mode is 10 because it appears twice, while all other values appear only once.

If we take the dataset: 5, 10, 10, 15, 15, 20, the modes are 10 and 15, as both appear twice. If all values appear only once, there is no mode.

Measures of Variability

Measures of variability describe the spread or dispersion of a dataset. The most common measures are the range, variance, and standard deviation.

Range

The range is the difference between the highest and lowest values in a dataset. To calculate the range, follow these steps:

1. Identify the maximum value.
2. Identify the minimum value.
3. Subtract the minimum value from the maximum value.

For example, in the dataset: 5, 10, 15, 20, 25, the maximum value is 25 and the minimum value is 5:

Range = 25 - 5 = 20

The range is 20.

Variance

Variance measures how far each number in a dataset is from the mean. It is calculated by following these steps:

1. Calculate the mean of the dataset.
2. Subtract the mean from each value to find the deviation.
3. Square each deviation.
4. Calculate the average of these squared deviations.

For example, consider the dataset: 5, 10, 15, 20, 25.

Step 1: Calculate the mean (as calculated earlier): 15.

Step 2: Find the deviations:

5 - 15 = -10

10 - 15 = -5

15 - 15 = 0

20 - 15 = 5

25 - 15 = 10

Step 3: Square each deviation:

(-10)² = 100

(-5)² = 25

(0)² = 0

(5)² = 25

(10)² = 100

Step 4: Average the squared deviations:

(100 + 25 + 0 + 25 + 100) ÷ 5 = 50

The variance is 50.

Standard Deviation

The standard deviation is the square root of the variance. It provides a measure of variability in the same units as the original data. To calculate the standard deviation, follow these steps:

1. Calculate the variance.
2. Take the square root of the variance.

Using the previous example where the variance is 50:

Standard Deviation = √50 ≈ 7.07

The standard deviation is approximately 7.07.

Visual Representation of Data

Visual representations of data help to illustrate the findings from descriptive statistics. Common methods include histograms, bar charts, and box plots.

Histograms

A histogram is a graphical representation of the distribution of numerical data. It consists of bars that represent the frequency of data points within certain ranges (bins).

To create a histogram, follow these steps:

1. Determine the range of the data.
2. Decide on the number of bins.
3. Count the number of data points in each bin.
4. Draw the bars for each bin.

Bar Charts

Bar charts are used for categorical data. Each category is represented by a bar, and the height of the bar represents the frequency of that category.

To create a bar chart, follow these steps:

1. Identify the categories.
2. Count the frequency of each category.
3. Draw bars for each category.

Box Plots

A box plot (or whisker plot) provides a visual summary of the central tendency, variability, and skewness of the data. It displays the median, quartiles, and potential outliers.

To create a box plot, follow these steps:

1. Calculate the median, first quartile (Q1), and third quartile (Q3).
2. Determine the interquartile range (IQR = Q3 - Q1).
3. Identify any outliers (values that fall below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR).
4. Draw a box from Q1 to Q3, with a line at the median.
5. Extend the whiskers to the smallest and largest values that are not outliers.

Summary

• Descriptive statistics summarise and present data effectively.
• Measures of central tendency include mean, median, and mode.
• Measures of variability include range, variance, and standard deviation.
• Visual representations of data include histograms, bar charts, and box plots.

Check your understanding

1. What is the mean of the dataset: 8, 12, 16, 20, 24?
2. How do you calculate the median of the dataset: 3, 7, 9, 11, 15, 19?
3. What is the range of the dataset: 2, 5, 8, 12, 15?
4. Explain the difference between variance and standard deviation.