Math Problem Statement

find standard deviation

Solution

To calculate the standard deviation of the data, we'll follow these steps:

  1. Calculate the mean μX\mu_X (already found as 1.607).

  2. Calculate each squared deviation from the mean, which is (xiμX)2(x_i - \mu_X)^2, and then multiply by the corresponding frequency fif_i.

  3. Sum the squared deviations.

  4. Divide by the total number of observations NN to find the variance.

  5. Take the square root of the variance to find the standard deviation.

Let's proceed with the calculation:

Step 1: Squared Deviations

(xiμX)2(x_i - \mu_X)^2

\hline x_i & f_i & (x_i - \mu_X) & (x_i - \mu_X)^2 \times f_i \\ \hline 0 & 24 & 0 - 1.607 = -1.607 & (-1.607)^2 \times 24 \approx 62.020 \\ 1 & 434 & 1 - 1.607 = -0.607 & (-0.607)^2 \times 434 \approx 159.823 \\ 2 & 261 & 2 - 1.607 = 0.393 & (0.393)^2 \times 261 \approx 40.237 \\ 3 & 79 & 3 - 1.607 = 1.393 & (1.393)^2 \times 79 \approx 153.476 \\ 4 & 19 & 4 - 1.607 = 2.393 & (2.393)^2 \times 19 \approx 108.948 \\ 5 & 13 & 5 - 1.607 = 3.393 & (3.393)^2 \times 13 \approx 149.497 \\ \hline \end{array}$$ ### Step 2: Sum the Squared Deviations Sum of squared deviations: $$62.020 + 159.823 + 40.237 + 153.476 + 108.948 + 149.497 \approx 674.001$$ ### Step 3: Calculate the Variance $$\text{Variance} = \frac{674.001}{830} \approx 0.812$$ ### Step 4: Calculate the Standard Deviation $$\text{Standard Deviation} = \sqrt{0.812} \approx 0.901$$ So, the standard deviation of the dataset is approximately **0.901**. Would you like further details or have any questions? --- Here are 8 related questions for further exploration: 1. How does the standard deviation help in understanding the distribution of the data? 2. What would the standard deviation be if all values were closer to the mean? 3. Can you calculate the variance manually from the standard deviation? 4. How does increasing the frequency of one of the days affect the standard deviation? 5. What is the interpretation of a low standard deviation in this context? 6. How would the standard deviation change if the highest frequency day (1 day) had an even higher frequency? 7. Can you plot the data distribution and analyze it visually? 8. How does the standard deviation compare to the mean in this dataset? **Tip:** The standard deviation is a measure of spread that tells us how much the values in a dataset typically differ from the mean. A smaller standard deviation means that the data points are closer to the mean.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Standard Deviation
Variance

Formulas

Standard Deviation formula
Variance formula

Theorems

-

Suitable Grade Level

Undergraduate level