Question

3 select all the statements that are true about standard deviation. a it is a measure of center. b it is a measure of variability. c it is the same as the mad. d it is calculated using the mean. e it is calculated using the median. 4 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:

Question 3:

• Option A: Standard deviation is a measure of variability (how spread out data is), not a measure of center (like mean/median). So A is false.
• Option B: Standard deviation measures the variability of data around the mean, so B is true.
• Option C: MAD (Mean Absolute Deviation) and standard deviation are different measures of variability, so C is false.
• Option D: Standard deviation is calculated using the mean (finding squared differences from the mean, etc.), so D is true.
• Option E: Standard deviation uses the mean, not the median, so E is false.

Step1: Sum the data values

The data is \(1, 2, 3, 4, 4, 5, 5, 6, 7, 8\). Sum them: \(1 + 2 + 3 + 4 + 4 + 5 + 5 + 6 + 7 + 8\)
\(= (1+2+3)+(4+4)+(5+5)+(6+7+8)\)
\(= 6 + 8 + 10 + 21 = 45\)

Step2: Divide by number of data points

There are \(n = 10\) data points. Mean \(\bar{x}=\frac{45}{10}=4.5\)

Step1: Find deviations from the mean

Mean \(\bar{x} = 4.5\). For each data point \(x_i\), calculate \(x_i - \bar{x}\):
\(1 - 4.5 = -3.5\), \(2 - 4.5 = -2.5\), \(3 - 4.5 = -1.5\), \(4 - 4.5 = -0.5\), \(4 - 4.5 = -0.5\),
\(5 - 4.5 = 0.5\), \(5 - 4.5 = 0.5\), \(6 - 4.5 = 1.5\), \(7 - 4.5 = 2.5\), \(8 - 4.5 = 3.5\)

Step2: Square the deviations

\((-3.5)^2 = 12.25\), \((-2.5)^2 = 6.25\), \((-1.5)^2 = 2.25\), \((-0.5)^2 = 0.25\), \((-0.5)^2 = 0.25\),
\((0.5)^2 = 0.25\), \((0.5)^2 = 0.25\), \((1.5)^2 = 2.25\), \((2.5)^2 = 6.25\), \((3.5)^2 = 12.25\)

Step3: Find the mean of squared deviations (variance)

Sum of squared deviations: \(12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25\)
\(= (12.25\times2)+(6.25\times2)+(2.25\times2)+(0.25\times4)\)
\(= 24.5 + 12.5 + 4.5 + 1 = 42.5\)
Variance \(s^2=\frac{42.5}{10}=4.25\)

Step4: Take square root of variance (standard deviation)

Standard deviation \(s = \sqrt{4.25} \approx 2.06\) (or exact form \(\sqrt{\frac{17}{4}}=\frac{\sqrt{17}}{2}\approx2.06\))

Answer:

B. It is a measure of variability, D. It is calculated using the mean

Question 4a: