Question

In this discussion, please: Must be 500 words

1. Think of an observation from daily or work life which you could quantify with one or more measures of central tendency (the mean, median, or mode)
   
   1. Explain and define the variable
      
      1. Explain the value of calculating a mean, median, or mode for that variable
         

If you used that measure of central tendency as your primary way of thinking about that phenomena, how useful would that be, how misleading might it be, and what other information might you want to know about that phenomena?

Answer 1

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. The mean, median and mode are all valid measures of central tendency.

Mean: The average number; found by adding all data points and dividing by the number of data points.

x̄ = ∑ x / n

the mean is not often one of the actual values that you have observed in your data set. However, one of its important properties is that it minimises error in the prediction of any one value in your data set. An important property of the mean is that it includes every value in your data set as part of the calculation.

Median: The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers). The median is less affected by outliers and skewed data.

Mode: The most frequent number—that is, the number that occurs the highest number of times. Normally, the mode is used for categorical data where we wish to know which is the most common category.

Example:

Let’s take an example to understand all of them better. Suppose we have data given below:

5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12

Mean:

x̄ = ∑ x / n

x̄ = (5 + 6 + 9 + 10 + 15 + 10 + 14 + 12 + 10 + 13 + 13 + 9 + 8 + 10 + 12) / 15

x̄ = 156 / 15

x̄ = 10.4

Median:

Observations in the ascending order are:
5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15

Here, n=15 is odd.

M= value of [(n+1)/2]th observation

= value of [(15+1)/2]th observation

= value of 8th observation

M=10

Mode:
In the given data, the observation 10 occurs maximum number of times (4)
Z = 10

Sample Variance:     

x	( x - x̄ )	( x - x̄ )²
5	-5.4	29.16
6	-4.4	19.36
9	-1.4	1.96
10	-0.4	0.16
15	4.6	21.16
10	-0.4	0.16
14	3.6	12.96
12	1.6	2.56
10	-0.4	0.16
13	2.6	6.76
13	2.6	6.76
9	-1.4	1.96
8	-2.4	5.76
10	-0.4	0.16
12	1.6	2.56
∑x = 156	∑ ( x - x̄ ) = 0	∑ ( x -x̄ )² = 111.6

Sample variance = [ ∑( x -x̄ )² / n]

                              = 111.6 / 15

                               = 7.44

Sample standard deviation = √[ ∑( x -x̄ )² /(n-)]

                                                 = √111.6/14

                                                 = √6.564

                                                  = 2.56