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Statistics " Measurements of central location chapter 2"
- 1. GROUP 2 PRESENTATION GROUP2 BBC2 STATISTICS 25 AUGUST 2022
- 2. STATISTICS MEASURE OF CENTRAL LOCATION
- 3. History and Definition Measures of central location was first found in the mid 1690’s in the writings of Edmund Halley. It has been used to summarize observations of a variable since the time of Galileo (1564 – 1642). Measures of central location are a combination of two words i.e., “measure” and “ central location”. Measure means methods and central location means average value of any statistical series. Central location is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution.
- 4. cont’d The measures of central tendency include: 1. The mode 2. The mean 3. The range 4. Upper and lower quartile 5. Variants 6. Median
- 5. THE MODE The mode is the most occurring value in a distribution. On a bar chart, the mode is the highest bar. If the data have multiple values that are tied for occurring the most frequently, you have a multi modal ( having several modes or maxima) distribution. Below is the example of mode from a data set showing the age of pupils at Chipiloni primary school. 6,6,6,7,7,7,7,8,8,8,9,9,9,9,9,10,11,11,12,12,13,13,13,13,14,14 The numbers have to be placed in order as shown above; it may either starts from the lowest to the highest or from the highest to the lowest. And then count how many times each number appears in a set. The one that appears the most is the mode. From the data set above, the most appearing number is 9, therefore the mode is 9. The value nine is appearing (frequency) five times greater than any other number
- 6. THE MEAN This is the arithmetic average and it is probably the measure of central tendency that you are most familiar. To calculate the mean, you add up all the total values given in a data sheet and divide the sum by the total number of values. Mean = sum of given data/total number of data Below is an example of test scores in a class: 39, 40,40,55, 62, 71, 71, 71, 86,95. 39+40+40+55+62+71+71+71+86+95 10 Mean= 63
- 7. THE MEDIAN Another common measure of central location is the median. It is useful when the data is skewed (sudden change in direction or position). Median means middle and the median is the middle of a set of data that has been into rank order. Median is the measure of central tendency which gives the value of the middle most observation in the data. Median of ungrouped data, we arrange the data values of the observations in ascending order and count the total number of observations. The number that is at the middle after arranging the values in ascending order is the median. E.g. 1,3,2,4,8,7,9,6 1. Arrange the numbers in order 1,2,3,4,5,6,7,8,9 The median is the number that is at the middle of the whole numbers, that is 5. When the values are even, we add the two middle number and divide them by 2 E.g. 1,2,3,4,5,6,7,8,9,10 The middle numbers are 5 and 6. Therefore, the median is 5+6/2= 5.5
- 8. RANGE The range of set of data is the difference between the largest and smallest values. The answer is found by subtracting the sample maximum and minimum. It is expressed in the same units as the data. Range of (y) = Max (y) – Min (y) E.g. find the range of the following data set. 10,15,20,25,30,35,40 Range = 40-10 Range = 30
- 9. UPPER AND LOWER QUARTILE The upper quartile or the third quartile (Q3), is the value under which 75% of data points are found when arranged in increasing order. The lower quartile or the first quartile (Q1), is the value under which 25% of data points are arranged in increasing order. How to calculate Quartiles 1. Order the data set from lowest to highest values. 2. Find the median. This is the second quartile Q2. 3. At Q2, split the ordered data set into two halves. 4. The lower quartile, Q1, is the median of the lower half of the data. 5. The upper quartile, Q3, is the median of the upper half of the data
- 10. Quartile cont’d The class has the following frequency distribution of marks scored by students 20, 30, 35, 45, 50, 55, 60, 65,88. find the first and third quartile. I. The numbers are already arranged in order. II. The median (Q2) is 50 To find first quartile: Q1= 30+35 2 Q1= 32.5 To find third quartile: Q3= 60+65 2 Q3= 62.5
- 11. REMARKS AND CONCLUSIONS Above scenarios shows that the measure of central location summarizes a list of numbers by a typical value and the three most common measures are: 1) The mean 2) The mode 3) The median. The median is the most appropriate measure of location for ordinal variable. However, mean is generally considered the best measure of central tendency and the most frequently used. This is the case because it uses all values in the data set to give you an average. Median is better than mean when measuring skewed data since it is not influenced by extremely large values.