Question

from unit 1,lesson 12 the number of different species of plants in some gardens is recorded. 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

The data set is \(1,2,3,4,4,5,5,6,7,8\). The sum \(S=\sum_{i = 1}^{n}x_{i}=1 + 2+3+4+4+5+5+6+7+8=45\).

Step2: Calculate the mean

The number of data - points \(n = 10\). The mean \(\bar{x}=\frac{S}{n}=\frac{45}{10}=4.5\).

Step3: Calculate the squared - differences

For each data - point \(x_{i}\), calculate \((x_{i}-\bar{x})^{2}\):
\((1 - 4.5)^{2}=(-3.5)^{2}=12.25\), \((2 - 4.5)^{2}=(-2.5)^{2}=6.25\), \((3 - 4.5)^{2}=(-1.5)^{2}=2.25\), \((4 - 4.5)^{2}=(-0.5)^{2}=0.25\), \((4 - 4.5)^{2}=(-0.5)^{2}=0.25\), \((5 - 4.5)^{2}=(0.5)^{2}=0.25\), \((5 - 4.5)^{2}=(0.5)^{2}=0.25\), \((6 - 4.5)^{2}=(1.5)^{2}=2.25\), \((7 - 4.5)^{2}=(2.5)^{2}=6.25\), \((8 - 4.5)^{2}=(3.5)^{2}=12.25\).
The sum of squared - differences \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=12.25+6.25 + 2.25+0.25+0.25+0.25+0.25+2.25+6.25+12.25 = 42.5\).

Step4: Calculate the variance

The variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\) (for a sample). Here \(n = 10\), so \(s^{2}=\frac{42.5}{9}\approx4.72\).

Step5: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}=\sqrt{\frac{42.5}{9}}\approx2.17\).

Answer:

a. \(4.5\)
b. Approximately \(2.17\)