Given the following frequency distribution, find the mean, median, mode and standard deviation. draw the histogram.

Question

Score          100     95     90     82

Frequency     1        2       3       4

Answer 1

The mode is the easiest since it's the score that occurs most often which is 82  

 

The median is also straightforward since it's the middle score when you sort them.  In sorted order the scores are 

82  82  82  82  90  90  90  95  95  100  

There are 10 scores so the median is the average of the 5th and 6th lowest, which are both equal to 90, so the median is 90  

 

To find the mean you add up all the scores and divide by the number of scores  

mean = (4*82 + 3*90 + 2*95 + 100)/10 = (328 + 270 + 190 + 100)/10 = 88.8  

 

To draw the histogram you plot the points (82,4), (90,3), (95,2) and (100,1).  You can use a bar chart  

 

The standard deviation is the most tedious to calculate.  The formula is 

s = √[ (1/(n-1)) * ∑(x_i - mean)² ]  

which means you have to find the difference between each data point and the mean, square it, add them all up, divide by the number of points minus one, and then take the square root of the result 

 

(82 - 88.8)² = 46.24    times 4 points at 82 = 184.96

(90 - 88.8)² = 1.44      times 3 points at 90 = 4.32

(95 - 88.8)² = 38.44    times 2 points at 95 = 76.88

(100 - 88.8)² = 125.44 times 1 point at 100 = 125.44  

 

So the total sum is 184.96 + 4.32 + 76.88 + 125.44 = 391.6

Divide by 9 to get 43.51111111

Take the square root to get ≅ 6.6