Question

use the display of data items to find the standard deviation.
the standard deviation is
(do not round until the final answer. then round to two decimal places as needed )

Explanation:

Step1: List out the data items and their frequencies

From the histogram, we have:

• \(x_1 = 6\), \(f_1=3\)
• \(x_2 = 9\), \(f_2 = 6\)
• \(x_3=12\), \(f_3 = 2\)

The total number of data items \(n=\sum_{i = 1}^{k}f_i=3 + 6+2=11\)

Step2: Calculate the mean \(\bar{x}\)

The formula for the mean of a frequency - distribution is \(\bar{x}=\frac{\sum_{i = 1}^{k}f_ix_i}{n}\)

\(\sum_{i = 1}^{k}f_ix_i=(3\times6)+(6\times9)+(2\times12)=18 + 54+24=96\)

\(\bar{x}=\frac{96}{11}\approx8.73\)

Step3: Calculate \(\sum_{i = 1}^{k}f_i(x_i-\bar{x})^2\)

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\(\sum_{i = 1}^{k}f_i(x_i-\bar{x})^2=22.3587+0.4374 + 21.3858=44.1819\)

Step4: Calculate the standard deviation \(s\)

The formula for the standard deviation of a sample (since we assume this is a sample from a larger population, and the formula for a population \(\sigma=\sqrt{\frac{\sum_{i = 1}^{k}f_i(x_i-\mu)^2}{N}}\) is similar in structure here with \(n\) used as \(N\)) is \(s=\sqrt{\frac{\sum_{i = 1}^{k}f_i(x_i-\bar{x})^2}{n - 1}}\) (for a sample) or \(s=\sqrt{\frac{\sum_{i=1}^{k}f_i(x_i - \bar{x})^2}{n}}\) (if it's a population. Here, we'll use the population formula as the problem doesn't specify sample/population. \(s=\sqrt{\frac{44.1819}{11}}\)

\(s=\sqrt{4.016536}\approx2.00\)

Answer:

\(2.00\)