Q. 3.208 - Elementary Statistics

Question

Chinchillidae Range. The family of mammals called Chinchillidae contains the chinchilla and viscachas, which are large South American rodents. The article. "Species Richness and Distribution of Neotropical Rodents, with Conservation Implications" (Mammalia Vol. $77$, Issue $1$. pp. $1-19$) by G. Amori et al reports the range sizes of each species of Chinchilladac. There are six species in the Chinchillidae family and the size of each species range, in square kilometers ($k{m}^2$) is given in the following table.

a. Obtain and interpret the population mean range of Chinchilladae species

b. Obtain and interpret the population standard deviation of the ranges of Chinchilladae species.

Step-by-Step Solution

Answer

(a) The mean is, $\mu \approx 524515.045$

(b) The standard deviation is, $\sigma \approx 709644.3094 k{m}^2$

Part (a) Step 1: Given information

We are given that,

Total no. of values, $n=6$

And the values are, 

$370954.56, 762822.31, 11213.32, 2070.38, 11943.02.1988086.68$

Part (a) Step 2: Simplify

We know that,

Mean is given as, 

$$\begin{gathered} \mu =\frac{Sum of values }{no. of values} \\ =\frac{370954.56+762822.31+11213.32+2070.38+11943.02+1988086.68}{6} \\ =\frac{3147090.27}{6}\approx 524515.045 \end{gathered}$$

Part (b) Step 1: Given information

We need to find out the standard deviation of the data.

Part (b) Step 2: Simplify

We know that,

Standard deviation is the square root of the variance and variance is the sum of squared deviations from the mean divided by $n$

Hence, Standard deviation is, $$\begin{gathered} \sigma =\sqrt{\frac{{\left(370954.56-524515.045\right)}^2+....+{\left(1988086.68-524515.045\right)}^2}{6}} \\ =\sqrt{\frac{{\left(-153560.485\right)}^2+...+{\left(1463571.635\right)}^2}{6}} \\ \approx 709644.3094 \end{gathered}$$