**Descriptive measures**

- The objective is to develop measures that can be used to summarize a data set.
- These descriptive measures are quantities whose values are determined by the data.

Most commonly used descriptive measures can be categorized as

**Measures of central tendency:**These are measures that indicate the most typical value or center of a data set.**Measures of dispersion:**These measures indicate the variability or spread of a dataset.

**The Mean**

- The most commonly used measure of central tendency is the mean.
- The mean of a data set is the sum of the observations divided by the number of observations.
- The mean is usually referred to as average.
- Arithmetic average; divide the sum of the values by the number of values (another typical value)
- For discrete observations:
- Sample mean: ̄x = x1 + x2 + . . . + xn/n
- Population mean: μ = x1 + x2 + . . . + xN / N

**Example**

- The marks obtained by ten students in an exam is 68, 79, 38, 68, 35, 70, 61, 47, 58, 66
- The sample mean is 68 + 79 + 38 + 68 + 35 + 70 + 61 + 47 + 58 + 66 / 10 = 590 /10 = 59

**Mean for grouped data: discrete single value data**

- The following data is the response from 15 individuals.
- 2, 1, 3, 4, 5, 2, 3, 3, 3, 4, 4, 1, 2, 3, 4
- x ̄ = f1x1 + f2x2 + . . . + fnxn / n

**Mean for grouped data: continuous data**

- x ̄ = f1m1 + f2m2 + . . . + fnmn / n

**Adding a constant**

- Let yi = xi + c where c is a constant then ̄y = ̄x + c
- Example: Recall the marks of students 68, 79, 38, 68, 35, 70, 61, 47, 58, 66.
- Suppose the teacher has decided to add 5 marks to each student.
- Then the data becomes 73, 84, 43, 73, 40, 75, 66, 52, 63, 71.
- The mean of the new data set is 640 10 = 64 = 59 + 5

**Multiplying a constant**

- Let yi = xi c where c is a constant then ̄y = ̄xc
- Example: Recall the marks of students 68, 79, 38, 68, 35, 70, 61, 47, 58, 66.
- Suppose the teacher has decided to scale down each mark by 40%, in other words each mark is multiplied by 0.4.
- Then the data becomes 27.2, 31.6, 15.2, 27.2, 14, 28, 24.4, 18.8, 23.2, 26.4
- The mean of the new data set is 236 10 = 23.6 = 59 × 0.4

Facebook Comments Box