Math Problem Statement

Value

18

19

15

20

25

31

17

35

27

22

34

29

40

33

21 find the standard deviation

Solution

To find the standard deviation (SD) of the given data set, follow these steps:

Given values:

18,19,15,20,25,31,17,35,27,22,34,29,40,33,2118, 19, 15, 20, 25, 31, 17, 35, 27, 22, 34, 29, 40, 33, 21

Step 1: Find the Mean (μ\mu)

The mean is calculated as:

μ=XN\mu = \frac{\sum X}{N}

where XX represents the values and NN is the number of values.

μ=18+19+15+20+25+31+17+35+27+22+34+29+40+33+2115\mu = \frac{18 + 19 + 15 + 20 + 25 + 31 + 17 + 35 + 27 + 22 + 34 + 29 + 40 + 33 + 21}{15}

μ=38515=25.67\mu = \frac{385}{15} = 25.67

Step 2: Find Each Value’s Deviation from the Mean and Square It

(Xμ)2(X - \mu)^2

We'll compute these in the next step.

Step 3: Compute the Variance (σ2\sigma^2)

The variance formula is:

σ2=(Xμ)2N\sigma^2 = \frac{\sum (X - \mu)^2}{N}

Let's calculate this.

Step 4: Compute the Standard Deviation (σ\sigma)

The standard deviation is:

σ=σ2\sigma = \sqrt{\sigma^2}

Now, let’s calculate the exact standard deviation. numpy as np

Given data set

data = np.array([18, 19, 15, 20, 25, 31, 17, 35, 27, 22, 34, 29, 40, 33, 21])

Compute standard deviation (population standard deviation)

std_dev = np.std(data, ddof=0) # Using population formula

std_devThe standard deviation of the given data set is approximately 7.38.

Would you like a detailed breakdown of the calculations?

Related Questions:

  1. What is the difference between population and sample standard deviation?
  2. How does standard deviation help in data analysis?
  3. Can standard deviation be negative? Why or why not?
  4. How does standard deviation relate to variance?
  5. What are real-life examples where standard deviation is useful?

Tip: A lower standard deviation means the data points are closer to the mean, while a higher standard deviation indicates more spread.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Deviation
Mean
Variance

Formulas

Mean (μ) = (ΣX) / N
Variance (σ²) = Σ(X - μ)² / N
Standard Deviation (σ) = √σ²

Theorems

Standard Deviation Theorem
Variance Theorem

Suitable Grade Level

Grades 9-12