Explanation:
Mean:
The mean is a measure of central tendency, also called the average. It is found as:
[tex]A ={\frac{1}{n} \sum_{i=1}^n (x_i)[/tex]
[tex]Where: \\ \\ n:\text{Total number of values} \\ \\ x_{1}:\text{the individual numbers in the data set.}[/tex]
Then:
[tex]A=\frac{78+95+77+92+87+79+94+86+88+94}{10}=\frac{870}{10} \\ \\ \boxed{A=87}[/tex]
Median:
The median is a measure of central tendency. It's the value of the center when the data set is sorted.
If the number of observations is odd the formula is given by:
[tex]x_m = x_\frac{n+1}{2}[/tex]
If the number of observations is even the formula is given by:
[tex]x_m = \frac{x_\frac{n}{2}+x_\frac{n+2}{2}}{2}[/tex]
Since the number of observations is even, we'll use the second formula. The sorted data set is:
[tex]77,78,79,86,87,88,92,94,94,95[/tex]
Therefore:
[tex]x_m = \frac{x_\frac{n}{2}+x_\frac{n+2}{2}}{2} \\ \\[/tex]
[tex]x_\frac{n}{2}}=x_{5}=87 \\ \\ x_\frac{n+2}{2}}=x_{6}=88 \\ \\ \\ x_m =\frac{87+88}{2} \\ \\ x_{m}=\frac{175}{2} \\ \\ \boxed{x_{m}=87.5}[/tex]
Range:
The Range is a measure of dispersion. It tells us by how much the values in the data set are likely to differ from their mean. It's found by subtracting the lowest from the highest value in the data set:
[tex]R=x_{M}-x_{m} \\ \\ \\ Where: \\ \\ R:Range \\ \\ x_{M}:Highest \ value \\ \\ x_{m}:Lowest \ value[/tex]
From the data set:
[tex]x_{m}= 77 \\ \\ x_{M}=95 \\ \\ \\ So: \\ \\ R=95-75 \\ \\ \boxed{R=18}[/tex]
Mean absolute deviation:
The mean deviation is a measure of dispersion. The formula is given by:
[tex]MAD =\frac{1}{n} \sum_{i=1}^n |x_i-\bar{x}|[/tex]
So:
[tex]\bar{x}:Mean \\ \\ x_{i}:each \ value \ in \ the \ data \ set[/tex]
The sorted data set:
[tex]77,78,79,86,87,88,92,94,94,95[/tex]
So;
[tex]|x_1-\bar{x}|= |77-87|=|-10|=10 \\ \\ |x_2-\bar{x}|= |78-87|=|-9|=9 \\ \\ |x_3-\bar{x}|= |79-87|=|-8|=8 \\ \\ |x_4-\bar{x}|= |86-87|=|-1|=1 \\ \\ |x_5-\bar{x}|= |87-87|=|0|=0 \\ \\ |x_6-\bar{x}|= |88-87|=|-10|=1 \\ \\ |x_7-\bar{x}|= |92-87|=5|=5 \\ \\ |x_8-\bar{x}|= |94-87|=|7|=7 \\ \\ |x_9-\bar{x}|= |94-87|=|7|=7 \\ \\ |x_10-\bar{x}|= |95-87|=|8|=8 \\ \\[/tex]
So:
[tex]MAD=\frac{1}{10}(10+9+8+1+0+1+5+7+7+8) \\ \\ \boxed{MAD=5.6}[/tex]
