Math Problem Statement

Observation 4,8,11,17,20,24,32 find standard deviation

Solution

To find the standard deviation of the data set {4,8,11,17,20,24,32}\{4, 8, 11, 17, 20, 24, 32\}, we will follow these steps:

Step 1: Calculate the Mean

The mean (average) of the data set is calculated as:

Mean=xin\text{Mean} = \frac{\sum x_i}{n}

Where:

  • xix_i are the observations,
  • nn is the number of observations.

Mean=4+8+11+17+20+24+327=116716.57\text{Mean} = \frac{4 + 8 + 11 + 17 + 20 + 24 + 32}{7} = \frac{116}{7} \approx 16.57

Step 2: Calculate Each Deviation from the Mean

Subtract the mean from each observation:

416.5712.57,816.578.57,1116.575.574 - 16.57 \approx -12.57,\quad 8 - 16.57 \approx -8.57,\quad 11 - 16.57 \approx -5.57 1716.570.43,2016.573.43,2416.577.43,3216.5715.4317 - 16.57 \approx 0.43,\quad 20 - 16.57 \approx 3.43,\quad 24 - 16.57 \approx 7.43,\quad 32 - 16.57 \approx 15.43

Step 3: Square Each Deviation

Square each of the deviations:

(12.57)2158.04,(8.57)273.43,(5.57)231.04(-12.57)^2 \approx 158.04,\quad (-8.57)^2 \approx 73.43,\quad (-5.57)^2 \approx 31.04 (0.43)20.18,(3.43)211.76,(7.43)255.22,(15.43)2238.06(0.43)^2 \approx 0.18,\quad (3.43)^2 \approx 11.76,\quad (7.43)^2 \approx 55.22,\quad (15.43)^2 \approx 238.06

Step 4: Calculate the Variance

The variance is the mean of the squared deviations:

Variance=(xiMean)2n\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n}

Variance=158.04+73.43+31.04+0.18+11.76+55.22+238.067567.73781.10\text{Variance} = \frac{158.04 + 73.43 + 31.04 + 0.18 + 11.76 + 55.22 + 238.06}{7} \approx \frac{567.73}{7} \approx 81.10

Step 5: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

Standard Deviation=81.109.00\text{Standard Deviation} = \sqrt{81.10} \approx 9.00

So, the standard deviation of the data set {4,8,11,17,20,24,32}\{4, 8, 11, 17, 20, 24, 32\} is approximately 9.00.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

  1. How does standard deviation help in understanding data distribution?
  2. What is the difference between variance and standard deviation?
  3. How do you calculate the standard deviation for a larger data set?
  4. How would removing an outlier affect the standard deviation of a data set?
  5. What is the significance of a low versus high standard deviation in a data set?

Tip: The standard deviation is a measure of the amount of variation or dispersion in a set of values. A lower standard deviation indicates that the values tend to be close to the mean, while a higher standard deviation indicates a wider range of values.

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Math Problem Analysis

Mathematical Concepts

Statistics
Descriptive Statistics
Standard Deviation

Formulas

Mean
Variance
Standard Deviation

Theorems

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Suitable Grade Level

Grades 11-12