Question

find the standard deviation for the group of data items.

the standard deviation is

(round to two decimal places as needed.)

Explanation:

Step1: Write out the data items

From the stem - and - leaf plot, the data items are \(x = \{25,32,35,42,45\}\). The number of data items \(n = 5\).

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

The mean formula is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
\(\sum_{i=1}^{5}x_{i}=25 + 32+35 + 42+45=179\).
\(\bar{x}=\frac{179}{5}=35.8\).

Step3: Calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\)

• For \(x_1 = 25\): \((25 - 35.8)^{2}=(- 10.8)^{2}=116.64\)
• For \(x_2 = 32\): \((32 - 35.8)^{2}=(-3.8)^{2}=14.44\)
• For \(x_3 = 35\): \((35 - 35.8)^{2}=(-0.8)^{2}=0.64\)
• For \(x_4 = 42\): \((42 - 35.8)^{2}=(6.2)^{2}=38.44\)
• For \(x_5 = 45\): \((45 - 35.8)^{2}=(9.2)^{2}=84.64\)

Step4: Calculate the variance \(s^{2}\)

The variance formula for a sample (assuming this is a sample, if it's a population, the formula is similar with \(n\) instead of \(n - 1\)). Here, we'll use the sample formula \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=116.64+14.44 + 0.64+38.44+84.64=254.8\).
\(s^{2}=\frac{254.8}{5 - 1}=\frac{254.8}{4}=63.7\).

Step5: Calculate the standard deviation \(s\)

The standard deviation formula is \(s=\sqrt{s^{2}}\).
\(s=\sqrt{63.7}\approx7.98\).

Answer:

\(7.98\)