Question

a study recorded the time it took for a sample of seven different species of frogs and toads eggs to hatch. the following table shows the times to hatch, in days. determine the range and sample standard deviation.
7 10 13 10 13 7 5
range = day(s)

Explanation:

Step1: Find maximum and minimum

The data set is \(7, 10, 13, 10, 13, 7, 5\). The maximum value \(max = 13\) and the minimum value \(min=5\).

Step2: Calculate the range

The formula for the range \(R\) of a data - set is \(R = max - min\). So \(R=13 - 5=8\).

Step3: Calculate the mean

The mean \(\bar{x}=\frac{7 + 10+13+10+13+7+5}{7}=\frac{65}{7}\approx9.286\).

Step4: Calculate the squared differences

\((7 - 9.286)^2\approx(- 2.286)^2 = 5.226\), \((10 - 9.286)^2\approx(0.714)^2 = 0.51\), \((13 - 9.286)^2\approx(3.714)^2 = 13.794\), \((10 - 9.286)^2\approx0.51\), \((13 - 9.286)^2\approx13.794\), \((7 - 9.286)^2\approx5.226\), \((5 - 9.286)^2\approx(-4.286)^2 = 18.37\).

Step5: Calculate the sum of squared differences

\(S=\sum_{i = 1}^{n}(x_i-\bar{x})^2=5.226+0.51 + 13.794+0.51+13.794+5.226+18.37=57.43\).

Step6: Calculate the sample variance

The formula for the sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n - 1}\), where \(n = 7\). So \(s^2=\frac{57.43}{6}\approx9.572\).

Step7: Calculate the sample standard deviation

The sample standard deviation \(s=\sqrt{s^2}=\sqrt{9.572}\approx3.1\).

Answer:

Range = 8 days, Sample standard deviation \(\approx3.1\) days