Determining measures of central tendency for grouped data
- 1. Measures of central tendency (Grouped data) Presented by: ALONA HALL
- 2. OBJECTIVES From this lesson, we hope to: Define the concept related to measures of central tendency Demonstrate how each measure of central tendency (mean, mode and median) is calculated from grouped data
- 3. The concept of measures of central tendency • The measures of central tendency are known as the mean, the mode and the median. They are all known as statistical averages but the arithmetic mean is the most popular average. • Each measure, when determined, gives the statistician/the observer information about what conclusions can be made about the set of data which it represents.
- 4. EXAMPLE The frequency table below shows the heights of 40 students in a high school a) Estimate the mean height b) Determine the modal class c) Estimate the median height of the set of students Height (cm) Frequency 135 – 139 5 140 – 144 4 145 – 149 2 150 – 154 12 155 – 159 15 160 – 164 2
- 5. The mean The mean (or x bar) is calculated using the following formula: 𝑥 = 𝑓𝑥 𝑓 (for grouped data) Where 𝑥 = 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 𝑓𝑥 = 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 × 𝑖𝑡𝑠 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 Since we have grouped data, that is, a range of numbers we need ONE value to represent the group. That value will be the MID-POINT 𝑖𝑠 𝑆𝑖𝑔𝑚𝑎 which simply means ‘find the sum of’
- 6. Example (cont’d) L.C.L means Lower Class Limit and U.C.L means Upper Class Limit 135 – 139 is a class interval in which the Lower Class Limit is 135 and the Upper Class Limit is 139. 140 – 144, …… 160 – 164 are also class intervals. Can you state which is the Lower or Upper Class Limit? Class Interval (cm) Mid-point (x) Frequency (f) fx 135 – 139 Class mid-point= 𝑡ℎ𝑒 𝐿𝐶𝐿 + 𝑡ℎ𝑒 𝑈𝐶𝐿 2 = 135+139 2 = 𝟏𝟑𝟕 5 137 x 5 = 685 140 – 144 You may calculate the second mid- point then use the pattern to determine the others 142 4 568 145 – 149 147 2 294 150 – 154 152 12 1824 155 – 159 157 15 2355 160 – 164 162 2 324 𝑓 = 40 𝑓𝑥 = 6050
- 7. Thus, applying the formula, 𝑥 = 𝑓𝑥 𝑓 = 6050 40 = 151.25 The estimated mean height is 151.25 cm It is an estimated mean because each group or class interval was represented by ONE number (the mid- point)
- 8. Now, let’s go to the mode The mode is the observation that appears most frequently. In this case we will not have one value as a mode. We will have a modal class. The class interval with the highest frequency is 155 - 159. Therefore modal class is 155 - 159.
- 9. Finally, let’s look at the median (denoted 𝑄2) This is the observation that appears in the centre after having arranged the values in ascending or descending order. In this case we will need to use a cumulative frequency table then estimate the median height from a cumulative frequency curve (Ogive)
- 10. Let’s determine the median Interval (cm) Cumulative Frequency ≤ 139.5 5 ≤ 144.5 5 + 4 = 9 ≤ 149.5 11 ≤ 154.5 23 ≤ 159.5 38 ≤ 164.5 40 We are adding or accumulating the frequencies considering the frequency, firstly, which would be less than the upper boundary of the first class interval (≤ 139.5) that is 5. Then we add the frequencies that are ≤ 144.5 which would be 5 + 4 = 9. We apply that procedure for the upper boundary of each class interval. The final frequency should correspond with the total number of observations (in this case, 40 )
- 11. How do we determine the upper class boundary (U.C.B)? 𝑇ℎ𝑒 𝑈. 𝐶. 𝐵 𝑜𝑓 𝑎 𝑐𝑙𝑎𝑠𝑠 = 𝑡ℎ𝑒 𝑈. 𝐶. 𝐿 𝑜𝑓 𝑡ℎ𝑎𝑡 𝑐𝑙𝑎𝑠𝑠 + 𝑡ℎ𝑒 𝐿. 𝐶. 𝐿 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑏𝑠𝑒𝑞𝑢𝑒𝑛𝑡 𝑐𝑙𝑎𝑠𝑠 2 For example, given that the first two class intervals are 135 – 139 and 140 – 144 then the U.C.B for the class interval 135 – 139 𝑈. 𝐶. 𝐵 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑙𝑎𝑠𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 139 + 140 2 = 139.5
- 12. We will now plot the Ogive. Bear in mind that we will plot the cumulative frequency against the upper boundary. The position of the median = 1 2 𝑛 since we are trying to locate the middle height then 1 2 × 40 = 20 𝑡ℎ 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 will suffice since we are estimating. NOTE: 1 2 𝑛 + 1 could also be used but will not make a big difference to the estimate. ALSO: The position of the median = 1 2 𝑛 when very large numbers are used.
- 13. The Cumulative Frequency curve (Ogive) has the characteristic shape seen in the diagram. We also break the horizontal axis in order to conveniently label the horizontal axis without starting at 0 through to 135.
- 14. The estimated median height = 153.5 cm . Remember to state the scale and label the axes and the general graph.