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 \(x_1 = 26\), \(x_2=33\), \(x_3 = 36\), \(x_4=43\), \(x_5 = 46\). The number of data items \(n = 5\).

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

The mean formula is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\).
\(\sum_{i=1}^{5}x_i=26 + 33+36 + 43+46=184\).
\(\bar{x}=\frac{184}{5}=36.8\).

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

• For \(x_1 = 26\): \((26 - 36.8)^2=(- 10.8)^2 = 116.64\)
• For \(x_2=33\): \((33 - 36.8)^2=(-3.8)^2 = 14.44\)
• For \(x_3 = 36\): \((36 - 36.8)^2=(-0.8)^2=0.64\)
• For \(x_4=43\): \((43 - 36.8)^2=(6.2)^2 = 38.44\)
• For \(x_5 = 46\): \((46 - 36.8)^2=(9.2)^2 = 84.64\)

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

The variance formula for a sample (also used when the population is not specified) is \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\).
\(\sum_{i = 1}^{5}(x_i-\bar{x})^2=116.64+14.44 + 0.64+38.44+84.64=254.8\).
\(s^2=\frac{254.8}{4}=63.7\).

Step5: Calculate the standard deviation \(s\)

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

Answer:

\(7.98\)