Question

Listed below are the numbers of heroic firefighters who lost their lives in the United States each year while fighting forest fires. The numbers are listed in order by year, starting with the year 2000. What important feature of the data is not revealed by any of the measures of variation?

Answer 1

Answer:

Step-by-step explanation:

Hello!

Data

Year/Firefighters

2000: 20

2001: 18

2002: 23

2003: 30

2004: 20

2005: 12

2006: 24

2007: 9

2008: 25

2009: 15

2010: 8

2011: 11

2012: 15

2013: 34

Common measures of variation

Variance

S²= $\frac{1}{n-1}[sumX^2-\frac{(sumX)^2}{n} ]$

∑X= 20 + 18 + 23 + 30 + 20 + 12 + 24 + 9 + 25 + 15 + 8 + 11 + 15 + 34

∑X= 264

∑X²= 20² + 18² + 23² + 30² + 20² + 12² + 24² + 9² + 25² + 15² + 8² + 11² + 15² + 34²

∑X²= 5770

n= 14

S²= $\frac{1}{13}[5770-\frac{(264)^2}{14} ]$

S²= 60.90

Standard deviation

The standard deviation is the square root of the variance

S= √S²= √60.90= 7.80

Coefficient of variation

Is a relative standard deviation, it is defined as the division of the standard deviation (S) by the mean (X[bar)]. It has not units and is usually expressed in percentage. It shows the variability in relation to the mean.

C.V.= $\frac{S}{X[bar]}$*100 = $\frac{7.80}{18.86}$*100= 41.36%

The mean or average is a measurement of position and gives you an idea of the central value of the distribution of the data.

X[bar]= ∑X/n= 264/14= 18.86

Range

In the interval between the max and min values. It allows you to have an idea of the dispersion of the values.

R= Xmax - Xmin= 34 - 8= 26

Xmax= 34

Xmin= 8

Inter Quartile Range

Is the difference between Quantile 1 (C₁) and Quantile 3 (C₃). It shows the 50% mid values of the sample.

IQR= C₃ - C₁= 23.5 - 11.5 = 12

Quantile 1 (C₁) is the value that leaves 25% of the sample below and 75% of the sample above.

Pos1: n/4= 14/4= 3.5

The first quantile is the number between position 3 and position 4

8, 9, 11, 12, 15, 15, 18, 20, 20, 23, 24, 25, 30, 34

C₁= (11+12)/2= 11.5

Quantile 3 (C₃) is the value that leaves 75% of the sample below and 25% of the sample above.

Pos3: n*3/4= 14*3/4= 10.5

The third Quantile is between position 10 and 11

8, 9, 11, 12, 15, 15, 18, 20, 20, 23, 24, 25, 30, 34

C₃= (23+24)/2= 23.5

The measurements of variation don't allow you to know the origin of the data, that it is from consecutive years.

I hope it helps!