Question

Mean of the 6 numbers 6,9,3,2,x ,y is 6 and the variance is 10. Find the value of ‘x’ & ‘y’ and obtain skewness and kurtosis to discuss its shape.

Answer 1

From mean: 6+9+3+2+x+y=6×6=36, so x+y=16.

From variance: (6-6)²+(9-6)²+(3-6)²+(2-6)²+(x-6)²+(y-6)²=6×10=60,

0+9+9+16+x²-12x+36+y²-12y+36=60,

x²-12x+y²-12y=60-106=-46,

x²+y²-12(x+y)=-46,

x²+y²-12×16=-46,

x²+y²=192-46=146.

x²+(16-x)²=146,

2x²-32x+256=146,

2x²-32x+110=0,

x²-16x+55=0=(x-5)(x-11).

So x=5 and y=11 or vice versa. These are the missing numbers.

The ordered dataset is 2, 3, 5, 6, 9, 11.

The median (2nd quartile Q2) is 5.5 which is slightly less than the mean, indicating a slightly positive skew (more/"heavier" data to the right).

This is caused by the value 11 which extends the right tail (kurtosis).

First quartile Q1=3, third quartile Q3=9 (left tail (whisker) is shorter than the right).

Without the data value 11, median=mean=5, variance=(9+4+0+1+16)/5=6, so inclusion of data value 11 increases the variance (data spread) to 10.