Measures of central tendency
A measure of central tendency is a single value that represents the centre of a data distribution. To determine the centre or central tendency of the data, we can calculate three measures: the mean, median, and mode. These are often described as ‘averages’, but each is calculated differently.
| Measure | Definition |
|---|---|
| Mean | The mean is the sum of all data points, divided by the total number of observations. |
| Median | The median is the midpoint of the data set. To derive the median, order the observations logically and then take the value at the centre. |
| Mode | The mode represents the most likely or most frequent outcome. |
Let’s see how to calculate each of these in more detail.
Calculating the mean
Let’s calculate the mean for the following data set.
[Data set = [89, 42, 12, 52, 33, 55, 97, 6, 61, 37, 83, 12, 92, 77, 58]]
Step 1: To calculate the mean, start by adding all the values together to calculate the total.
[Total = 89+42+12+52+33+55+97+6+61+37+83+12+92+77+58=806]
Step 2: Now, divide the total by the number of values. In this case, there are 15.
Mean = (frac{806}{15} = 53.73)
The mean of this data set is 53.73.
Calculating the median
Let’s calculate the median for the same data set as above.
[Data set = [89, 42, 12, 52, 33, 55, 97, 6, 61, 37, 83, 12, 92, 77, 58]]
Step 1: To calculate the median, order the data set from lowest to highest. The data becomes:
[[6, 12, 12, 33, 37, 42, 52, 55, 58, 61, 77, 83, 89, 92, 97]]
Step 2: If you have a small number of values, it’s easy to find the midpoint; but, for larger sets of values, you can use a formula to calculate where you’ll find the median.
Median position = (frac{(Number of values + 1)}{2})
Median position = (frac{(15 + 1)}{2})= (8)
Step 3: Here, the median of the data set is the number in the median position. In this case, the number in the eighth position is 55, which is the median of the data set.
Step 4: If your data set is even, calculating the median position won’t yield a whole number. For instance, for the data set given below:
(Data set = [6, 12, 12, 33, 37, 42, 52, 55, 58, 61, 77, 83, 89, 92])
Median position = (frac{(14+1)}{2}) = 7.5
In this instance, you’d take an average of the values in the seventh and eighth positions. In our example, these are 52 and 55 and by summing these values (52+55) and dividing by two, you have the median value: 53.5.
Calculate the mode
Calculating the mode involves finding the most common value in a data set.
Step 1: To calculate the mode, you need to count the number of occurrences, or frequency, of each number.
Step 2: Now, using the same data set as above, tally up the frequency of each number.
| Frequency | Value |
|---|---|
| 1 | 6 |
| 2 | 12 |
| 1 | 33 |
| 1 | 37 |
| 1 | 42 |
| 1 | 52 |
| 1 | 55 |
| 1 | 58 |
| 1 | 61 |
| 1 | 77 |
| 1 | 83 |
| 1 | 89 |
| 1 | 92 |
| 1 | 97 |
Step 3: As you can see, 12 is the only number that occurs more than once; therefore, 12 is the mode of this data set.
Task
- Practise calculating measures of central tendency. Round all your calculations to two decimal places.
- What are the mean, median, and mode for the data set: ([-14, 3, -8, 1, 2, 19, 14, 4, -8, -1, 7, 20])?
- Note that there is no repeating number in this data set. How will you calculate the mode?
- Share your responses to the question in the comments below.
Next, you’ll move on to learning about measures of spread.
Introduction to Data Science for Business
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