Question

Mr. Green teaches mathematics and his class recently finished a unit on statistics. The student scores on this unit are: 40 47 50 50 50 54 56 56 60 60 62 62 62 63 65 70 70 72 76 77 80 85 85 95 Perform a statistical analysis on this data set. a) Find the measures of central tendency and the range of the data.

Answer 1

Answer:

Mean = 64.46, Median = 62 and Mode = Bi-modal (50 and 62)

Range of the data is 55.

Step-by-step explanation:

We are given that Mr. Green teaches mathematics and his class recently finished a unit on statistics.

The student scores on this unit are:  40, 47, 50, 50, 50, 54, 56, 56, 60, 60, 62, 62, 62, 63, 65, 70, 70, 72, 76, 77, 80, 85, 85, 95.

We know that Measures of Central Tendency are: Mean, Median and Mode.

• Mean is calculated as;

                   Mean  =  $\frac{\sum X}{n}$

where  $\sum X$ = Sum of all values in the data

               n = Number of observations = 24

So, Mean  =  $\frac{40+ 47+ 50+ 50+ 50+ 54+ 56+ 56+ 60 +60+ 62+ 62+ 62+ 63+ 65+ 70+ 70+ 72+ 76+ 77+ 80+ 85+ 85+ 95}{24}$

=  $\frac{1547}{24}$  =  64.46

So, mean of data si 64.46.

For calculating Median, we have to observe that the number of observations (n) is even or odd, i.e.;

• If n is odd, then the formula for calculating median is given by;

                     Median  =  $(\frac{n+1}{2})^{th} \text{ obs.}$

• If n is even, then the formula for calculating median is given by;

                     Median  =  $\frac{(\frac{n}{2})^{th}\text{ obs.} +(\frac{n}{2}+1)^{th}\text{ obs.} }{2}$

Now here in our data, the number of observations is even, i.e. n = 24.

So, Median  =  $\frac{(\frac{n}{2})^{th}\text{ obs.} +(\frac{n}{2}+1)^{th}\text{ obs.} }{2}$

                    =  $\frac{(\frac{24}{2})^{th}\text{ obs.} +(\frac{24}{2}+1)^{th}\text{ obs.} }{2}$

                    =  $\frac{(12)^{th}\text{ obs.} +(13)^{th}\text{ obs.} }{2}$

                    =  $\frac{62 + 62 }{2}$  =  $\frac{124}{2}$  =  62

Hence, the median of the data is 62.

• A Mode is a value that appears maximum number of times in our data.

In our data, there are two values which appear maximum number of times, i.e. 50 and 62 as these both appear maximum 3 times in the data.

This means our data is Bi-modal with 50 and 62.

• The Range is calculated as the difference between the highest and lowest value in the data.

                      Range  =  Highest value - Lowest value

                                   =  95 - 40 = 55

Hence, range of the data is 55.