Question

the number of different species of plants in some gardens is reported: 1, 2, 3, 4, 4, 5, 5, 6, 7, 8. a. what is the mean? b. what is the standard deviation?

Explanation:

Step1: Calculate the sum of data

$1 + 2+3 + 4+4 + 5+5 + 6+7 + 8=45$

Step2: Calculate the number of data points

There are $n = 10$ data - points.

Step3: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{45}{10}=4.5$

Step4: Calculate the squared - differences from the mean

$(1 - 4.5)^2=(-3.5)^2 = 12.25$, $(2 - 4.5)^2=(-2.5)^2 = 6.25$, $(3 - 4.5)^2=(-1.5)^2 = 2.25$, $(4 - 4.5)^2=(-0.5)^2 = 0.25$, $(4 - 4.5)^2=(-0.5)^2 = 0.25$, $(5 - 4.5)^2=(0.5)^2 = 0.25$, $(5 - 4.5)^2=(0.5)^2 = 0.25$, $(6 - 4.5)^2=(1.5)^2 = 2.25$, $(7 - 4.5)^2=(2.5)^2 = 6.25$, $(8 - 4.5)^2=(3.5)^2 = 12.25$

Step5: Calculate the variance

The variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}=\frac{12.25 + 6.25+2.25 + 0.25+0.25 + 0.25+0.25 + 2.25+6.25 + 12.25}{10}=\frac{42.5}{10}=4.25$

Step6: Calculate the standard deviation

The standard deviation $s=\sqrt{s^{2}}=\sqrt{4.25}\approx2.06$

Answer:

a. $4.5$
b. $\approx2.06$