Question

{57, 53, 53, 71, 73, 57, 61, 58, 78. 64, 54, 69, 56, 58, 49, 56, 53, 52, 82, 62, 61, 60, 71, 75, 60} Whats the mean?. and the iqr? what is the five number summary? what is Q3? The Median is 60.

Answer 1

Given the data set:

$\lbrace57,53,53,71,73,57,61,58,78,64,54,69,56,58,49,56,53,52,82,62,61,60,71,75,60\rbrace$

• You can find the Mean by adding all the values and dividing the sum by the number of values in the data set:

$Mean=\frac{57+53+53+71+73+57+61+58+78+64+54+69+56+58+49+56+53+52+82+62+61+60+71+75+60}{25}$$Mean\approx61.72$

• By definition the term for the third quartile can be found with this formula:

$\frac{3}{4}(n+1)$

Where "n" is the number of observations.

In this case:

$n=25$

Then:

$\frac{3}{4}(25+1)\approx19.5$

Since it is an integer, you get that the position of the terms is:

$Q_3=\frac{69+71}{2}=70$

Because, when you order the data set, 69 is the 19th value and 71 is the 20th value. Then, the third quartile is the average between them:

$\lbrace49,52,53,53,53,54,56,56,57,57,58,58,60,60,61,61,62,64,69,71,71,73,75,78,82\rbrace$

• By definition:

$IQR=Q_3-Q_1$

And the term position of the first quartile is found with:

$\frac{n+1}{4}$

You get:

$\frac{25+1}{4}=6.5$

Therefore, you can determine that:

$Q_1=\frac{54+56}{2}=55$

Then:

$IQR=70-55=15$

• By definition, the Five-Number Summary is:

- The minimum value:

$Minimum=49$

- The first quartile:

$Q_1=55$

- The median:

$Median=60$

- The third quartile:

$Q_3=70$

- The maximum value:

$Maximum=82$

Hence, the answers are:

• Mean:

$Mean\approx61.72$

• IQR:

$IQR=15$

• Five-Number Summary:

$Minimum=49$$Q_1=55$$Median=60$

$Q_3=70$

$Maximum=82$

• Third quartile:

$Q_3=70$