Summarizing Data
Measuring Spread: the Range
When datasets are large, it's important to find a smaller set of values able to describe its overall behavior.
The first indicator is the range of the dataset, that we obtain by subtracting its lowest value from its highest value. The range describes how spread are data.
Example: The range of the set is
Measuring the Central Tendency: the Averages
The measures of central tendency (averages) summarize a dataset using one calculated value, that conveys information about the typical values of data and their distribution.
The most common measures of central tendency of a dataset are the (arithmetic) mean, the median and the mode.
Mean
The (arithmetic) mean is the sum of all values in the dataset, divided by their number.
If the dataset contains qualitative data, it's not possible to calculate the mean of these data.
Example 1: The mean of the set is
Example 2: The mean of the set is not defined.
Median
The median is the middle value of an ordered dataset, with data arranged in increasing order.
- If the ordered data set contains an odd number of values, the median will be the middle value in the list.
- If the ordered data set contains an even number of values, the median will be the (arithmetic) mean of the two middle values.
If the dataset contains qualitative data, it's not possible to calculate the median of these data.
Example 1: Calculate the median of the set .
- The set contains 11 values (odd)
- Sort values in increasing order:
- The median is , that is the middle element of the list above.
Example 2: Calculate the median of the set
- The set is already ordered, and contains 6 values (even)
- The values in the middle of the list are
- The median of the given set is the mean of these values:
Example 3: The median of the set is not defined.
Mode
The mode is the value with the highest frequency in a dataset (i.e. that occurs most often)
It is not necessarily unique, and it is defined also for qualitative datasets.
Example 1: The mode of the set is .
Example 2: The set has two modes: and .
Example 3: The mode of the set is .