The Arithmetic Mean.pptx

The Arithmetic Mean

1. The Arithmetic Mean

2. Perhaps the most important measure of location is the mean . The mean provides a measure of central location . The mean of a data set is the average of all the data values. The sample mean 𝒙 is the point estimator of the population mean µ. Mean

3. 𝑀𝑒𝑎𝑛 = The arithmetic mean is the most commonly used average. Definition: A value obtained by dividing the sum of all observations by their number, that is 𝑺𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒕𝒉𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔 The Arithmetic Mean

4. The population mean is a fixed quantity. The sample mean is a variable because different samples from the same population tend to have different mean. The Arithmetic Mean

5. The mean may correspond to either a population or a sample from the population. If the given set of observations represents a population, the mean is called population mean which is traditionally denoted by 𝝁 (the Greek letter mu). Thus the population mean of a set of 𝑵 observations 𝒙 𝟏 , 𝒙 𝟐 , 𝒙 𝟑 , … , 𝒙 𝑵 is given as 𝒙 𝟏 + 𝒙 𝟐 + 𝒙 𝟑 + ⋯ + 𝒙 𝑵 𝝁 = 𝑵 𝒙 𝑵 = 𝒊=𝟏 𝒊 𝑵 Where 𝛴 , the Greek capital Sigma, is a convenient symbol for summation. Ungrouped data Population mean The Arithmetic Mean

6. The Arithmetic Mean (Ungrouped Data) If the given set of observations represents a sample, the mean is called a sample mean , usually denoted by placing a bar over the symbol used to represent the observations or the variables. The mean of a set of n observations 𝒙 𝟏 , 𝒙 𝟐 , 𝒙 𝟑 , … , 𝒙 𝒏 is defined as: 𝒙 𝟏 + 𝒙 𝟐 + 𝒙 𝟑 + ⋯ + 𝒙 𝒏 𝒙 = 𝒏 = 𝒏 𝒙 𝒊 𝒊=𝟏 𝒏 Where 𝒙 is the mean of a sample of size n. Ungrouped data Sample mean

7. The Arithmetic Mean (Example of Population Mean ) Example: (Ungrouped Data) The number of employees at 5 different drugstores are 3, 5, 6, 4 and 6. Treating the data as population, find the mean number of employees for the 5 stores. Solution: Since the data are considered as finite population, 𝝁 = 3+5+6+4+ 𝟔 𝟓 = 𝟒. 𝟖 = 𝟓 𝒆𝒎𝒑𝒍𝒐𝒚𝒆𝒆𝒔

8. The Arithmetic Mean (Example of Sample Mean) Example : (Ungrouped Data) The sample of marks obtained by 9 students are given as 45,32,37, 46, 39, 41, 48, 36, 𝟑𝟔. Calculate the arithmetic mean. Solution: The mean is given by = 45 + 32 + 37 + 46 + 39 + 41 + 48 + 36 + 36 𝟗 = 𝟑𝟔𝟎 𝟗 = 𝟒𝟎 𝒎𝒂𝒓𝒌𝒔

9. Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a college town. The monthly rents for these apartments are listed below. Sample Mean 𝒙 545 715 530 690 535 700 560 700 540 715 540 540 540 625 525 545 675 545 550 550 565 550 625 550 550 560 535 560 565 580 550 570 590 572 575 575 600 580 670 565 700 585 680 570 590 600 649 600 600 580 670 615 550 545 625 635 575 650 580 610 610 675 590 535 700 535 545 535 530 540

10. 𝑥 𝑖 41,356 𝑥 = = = 𝑛 70 Sample Mean 𝒙 590.80 Example: Apartment Rents

11. The Arithmetic Mean (Grouped Data) To calculate the mean of grouped data, the first step is to determine the midpoint of each interval or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean. Mathematically, 𝒇 𝒊 𝒇 𝒊 𝒙 𝒊 𝒙 = , 𝒊 = 𝟏, 𝟐, 𝟑, … , 𝒏 Grouped data Sample mean

12. The Arithmetic Mean (Example of Sample Mean) Example : (Grouped Data) In Tim's office, there are 25 employees. Each employee travels to work every morning in his or her own car. The distribution of the driving times (in minutes) from home to work for the employees is shown in the table below. Calculate the mean of the driving times. Driving Times (minutes) Number of Employees to less than 10 3 10 to less than 20 10 20 to less than 30 6 30 to less than 40 4 40 to less than 50 2

13. The Arithmetic Mean (Example of Sample Mean) Solution : (Grouped Data) Driving Times (minutes) Number of Employees ( 𝒇 𝒊 ) Mid Points ( 𝒙 𝒊 ) 𝒇 𝒊 𝒙 𝒊 𝒇 𝒊 𝒙 𝒊 𝒙 = 𝒇 𝒊 𝟓𝟒𝟓 𝒙 = 𝟐𝟓 𝒙 = 𝟐𝟏. 𝟖 to less than 10 3 5 15 10 to less than 20 10 15 150 20 to less than 30 6 25 150 30 to less than 40 4 35 140 40 to less than 50 2 45 90 𝒇 𝒊 = 𝟐𝟓 𝒇 𝒊 𝒙 𝒊 = 𝟓𝟒𝟓

14. Properties of Arithmetic Mean Some important properties of the arithmetic mean are as follows: The sum of deviations of the items from their arithmetic mean is always zero, i.e. (𝒙 – 𝒙) = 𝟎. The sum of the squared deviations of the items from Arithmetic Mean (A.M) is minimum, which is less than the sum of the squared deviations of the items from any other values. If we increase or decrease every value of the data set by a specified weight, then the mean is also increased/decreased by the same digit. If we multiplied or divided every value of the data set by a specified weight, then the mean is also multiplied/divided by the same digit.

15. Advantages of Arithmetic Mean Arithmetic mean is simple to understand and easy to calculate. It is rigidly defined by a mathematical formula. It is suitable for further algebraic treatment. It is least affected fluctuation of sampling. It takes into account all the values in the series.

16. Disadvantages of Arithmetic Mean It is highly affected by the presence of a few abnormally high or abnormally low scores. In absence of a single item, its value becomes inaccurate. It can not be determined by inspection. It gives sometimes fallacious conclusions. In a highly skewed distribution, the mean is not an appropriate measure of average. If the grouped data have “open end” classes, mean can't be calculated without assuming the limits.

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