The Interquartile Range equation is Q3-Q1

First, make sure the data is in ascending order. Then split the data up so that it each quartile has 25% of the data, or think of it as splitting the data into 4 equal parts.

• $\displaystyle Q_{1}$ is the "middle" value in the first half of the rank-ordered data set.
• $\displaystyle Q_{2}$ is the median value in the overall set.
• $\displaystyle Q_{3}$ is the "middle" value in the second half of the rank-ordered data set.

$\displaystyle Q_{1}=2$

$\displaystyle Q_{2}=4$

$\displaystyle Q_{3}=6$

$\displaystyle Q_{3}-Q_{1}= 6-2=4$

Standard deviation is essentially the average distance from the mean of a group of numbers.  There are a number of steps in computing standard deviation, but the steps are not too complicated if you take them one at a time.  First, find the mean of the values.  Second, subtract the mean from the first value and square the result.  Do this for all remaining values.  Third, add these results together.  Fourth, divide this value by the number of values.  Finally, find the square root of the result. 

1: $\displaystyle 50/4=12.5$

2: $\displaystyle (15-12.5)^{2}+(12-12.5)^{2}+(10-12.5)^{2}+(13-12.5)^{2}$

3: $\displaystyle 6.25+0.25+6.25+0.25=13$

4: $\displaystyle 13\div 4=3.25$

5: $\displaystyle \sqrt{3.25}=1.8$

There are five steps to finding the standard deviation of a group of values. First, find the mean of the values.  Second, subtract the mean from the first value and square the result.  Do this for all remaining values.  Third, add these results together.  Fourth, divide this value by the number of values minus one.  Finally, find the square root of the result. 

Variation measures the average difference between values within a group.  The process is not complicated but there are four steps that can take time.  First, find the mean of the values.  Second, subtract the mean from the first value and square the result.  Do this for all remaining values.  Third, add these results together.  Fourth, divide this value by the number of values in the group minus one (in this case, there are four days). 

1: $\displaystyle 50/4=12.5$

2: $\displaystyle (15-12.5)^{2}+(12-12.5)^{2}+(10-12.5)^{2}+(13-12.5)^{2}$

3: $\displaystyle 6.25+0.25+6.25+0.25=13$

4: $\displaystyle 13/(4-1)=4.33$

Note that to find the standard deviation, we would simply take one additional step of finding the square root of the variance. 

There are four steps to finding the variance of values within a group.  First, find the mean of the values.  Second, subtract the mean from the first value and square the result.  Do this for all remaining values.  Third, add these results together.  Fourth, divide the result  by the number of values in the group minus one (in this case, there are seven days, so you must divide by six).  

To find the mean of the whole population, multiply the female's average by the number of females, and then multiply the male's average by the number of males. Sum up these products and divide by the total number of males AND females:

 $\displaystyle \left [ \left ( 65\times 18\ females\right )+\left ( 70\times 15\ males\right ) \right ]\times\frac{1}{33\ total\ people} = \frac{2220}{33} = 67.3\ inches$

This question illustrates the close relationship between the concepts of variance and standard deviation.  We can find variance even though we do not know the values in the population if we know the standard deviation.  Simply square the standard deviation to find the variance. 

$\displaystyle 7.5^{2}=56.25$

The $\displaystyle \small 90^{th}$-percentile is the value such that $\displaystyle \small 90$ percent of values are less than it.

Using a normal table or calculator (such as R, using the command qnorm(0.9)), we get that the approximate $\displaystyle \small 90^{th}$-percentile is about $\displaystyle \small 1.28$.

In order to find the first and third quartiles, we haave to find the 25th and 75th percentiles, respectively.

To find the $\displaystyle n^{th}$ percentile, we find the product of $\displaystyle 0.n$ and the number of items $\displaystyle k$ in the set.

$\displaystyle p=0.n*k$

We then round that number $\displaystyle p$ up if it is not a whole number, and the $\displaystyle p^{th}$ term in the set is the $\displaystyle n^{th}$ percentile.

For this problem, to find the $\displaystyle 25^{th}$ and $\displaystyle 75^{th}$ percentile, we first find that there are 14 items in the set. We find their respective products to be

$\displaystyle p_{25}=0.25*14=3.5\approx4$    and 

$\displaystyle p_{75}=0.75*14=10.5\approx11$

As such, the $\displaystyle 25^{th}$ and $\displaystyle 75^{th}$ percentiles are the fourth and eleventh terms in the set, or 

$\displaystyle Q_1=12,\, Q_3=29$

The interquartile range of a set of scores is the difference between the third and first quartile - that is, the difference between the 75th and 25th percentiles. The 75th percentile is between 78 and 86, so, if 41 is subtracted from those numbers, the upper and lower bounds of the 25th percentile can be found.

$\displaystyle P_{70} < P_{75} < P_{80}$

$\displaystyle P_{70}- IR < P_{75} - IR < P_{80}- IR$

$\displaystyle P_{70}- IR < P_{25} < P_{80}- IR$

$\displaystyle 78- 41 < P_{25} < 86- 41$

$\displaystyle 37 < P_{25} < 45$

Of our choices, only 40 falls in this range.