Measures of central tendency in data sets

Descriptive statistics help summarise and interpret data by identifying typical or average values within a set of results. Measures of central tendency are mathematical methods used to find a representative value that describes the centre of a data set.

Types of central tendency measures

• Mode - This is the value or score that appears most frequently in a data set. A data set can have more than one mode if multiple values occur with the same highest frequency.
• Mean - Often referred to as the average, the mean is calculated by summing all the values in a data set and then dividing by the total number of values.
• Median - This is the middle value in a data set when the values are arranged in ascending or descending order.

Example calculation for central tendency

Consider a data set of test scores: 4, 6, 7, 7, 9, 11, 14.

• Mode - The score 7 appears twice, more than any other value, so the mode is 7.
• Mean - Add all scores (4 + 6 + 7 + 7 + 9 + 11 + 14 = 58) and divide by the number of scores (7). So, 58 ÷ 7 = 8.29 (rounded to two decimal places).
• Median - Arrange in order (already done) and find the middle value. With 7 scores, the 4th value is 7, so the median is 7.

Measures of dispersion in data sets

While central tendency describes the average, measures of dispersion focus on the spread or variability of data points within a set. These measures indicate how much the data values differ from each other or from the central value.

Types of dispersion measures

• Range - This shows the difference between the highest and lowest values in a data set. It is calculated by subtracting the smallest value from the largest and adding 1 to account for the inclusive range.
• Variance - This measures the average of the squared differences between each data point and the mean.
• Standard deviation - A widely used measure of spread, it is the square root of the variance. It indicates how much, on average, data points differ from the mean.

Example calculation for dispersion

Using the same data set: 4, 6, 7, 7, 9, 11, 14.

• Range - Largest value (14) minus smallest value (4) plus 1 = 14 - 4 + 1 = 11.
• Variance - First, find the mean (8.29). Then calculate the squared differences from the mean for each value, sum them, and divide by the number of values. This results in a variance of approximately 10.24 (rounded).
• Standard deviation - Take the square root of the variance (√10.24 ≈ 3.20, rounded to two decimal places).

Key numerical representations

Beyond central tendency and dispersion, data can be expressed in different numerical formats to simplify comparisons or highlight proportions.

Common numerical formats

• Ratio - This expresses a relationship between two or more quantities, showing how many times one value contains or is contained within another.
• Percentage - This represents a fraction of 100, used to show proportions out of a whole. It is denoted by the % symbol.
• Fraction - This shows a part of a whole, written as one number over another. The top number (numerator) is divided by the bottom number (denominator).

Example of numerical representations

For a survey of 200 people where 50 prefer tea over coffee:

• Ratio - The ratio of tea to coffee preference could be 50:150, simplified to 1:3.
• Percentage - The percentage preferring tea is (50 ÷ 200) × 100 = 25%.
• Fraction - The fraction preferring tea is 50/200, simplified to 1/4.

Different types of data presentation methods

Presenting data visually helps to identify patterns, trends, and relationships more easily. Various graphical methods are used to display statistical information in an accessible and clear manner.

Methods of visual data presentation

• Frequency table (tally chart) - A table that lists categories of data with marks or symbols added each time a value is observed.
• Line graph - A chart that connects data points with lines to illustrate changes or trends over time or across categories.
• Pie chart - A circular graph divided into segments, each representing a proportion of the whole.
• Bar chart - A graph using rectangular bars (horizontal or vertical) to represent the size of different categories. Bars are separated to distinguish individual values.
• Histogram - A chart similar to a bar chart but used for continuous data, with bars touching each other to show the distribution across grouped ranges.
• Scatter diagram - A plot of individual points to display the relationship between two variables.