Math Problem Statement

Seth asked a random sample of 20 students how much cash they had in their pocket. The following frequency table gives the responses. cash($) freq 0. 1

  1.    2
    
  2.   4
    
  3. 11
    
  4. 1
    
  5. 1 Find the standard deviation. Give your answer to at least 2 decimal places

Solution

To calculate the standard deviation of a frequency distribution, we use the formula:

σ=f(xμ)2N\sigma = \sqrt{\frac{\sum{f(x - \mu)^2}}{N}}

Where:

  • xx = the cash amount (given in the table)
  • ff = frequency (number of students)
  • NN = total number of students
  • μ\mu = mean of the distribution

Let's follow these steps:

  1. Find the mean μ\mu: μ=fxN\mu = \frac{\sum{f \cdot x}}{N}

  2. Use the mean to calculate the variance: Variance=f(xμ)2N\text{Variance} = \frac{\sum{f(x - \mu)^2}}{N}

  3. Standard deviation is the square root of the variance.

I'll now compute this.The standard deviation of the cash amounts is approximately 211.65211.65.

Would you like further details or have any questions about the process?

Here are 5 related questions to expand on this concept:

  1. How would the standard deviation change if the extreme outlier ($985) was removed?
  2. Can you explain why the mean and standard deviation are sensitive to outliers?
  3. What would happen to the standard deviation if all the cash amounts were doubled?
  4. How is standard deviation different from variance, and why is it often preferred?
  5. Could you compare this standard deviation with that of a larger sample?

Tip: When calculating the standard deviation, extreme outliers can significantly affect the result, so always check the data for such values!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Standard Deviation
Frequency Distribution
Mean
Variance

Formulas

Mean formula: μ = (Σf * x) / N
Variance formula: Variance = Σf * (x - μ)^2 / N
Standard deviation formula: σ = √(Variance)

Theorems

-

Suitable Grade Level

Grades 9-12