Question

1. 10, 15, 25, 35, 50, 75

mean =
range =
variance =
standard deviation
\t\tn\tm\tn - m\t(n - m)²

Explanation:

Step1: Calculate the Mean (m)

The mean is the sum of all values divided by the number of values. The data set is \(10, 15, 25, 35, 50, 75\). First, find the sum: \(10 + 15 + 25 + 35 + 50 + 75 = 210\). There are \(n = 6\) values. So, the mean \(m=\frac{210}{6}=35\).

Step2: Calculate the Range

The range is the difference between the maximum and minimum values. The maximum value is \(75\) and the minimum is \(10\). So, range \(= 75 - 10 = 65\).

Step3: Calculate the Deviations and Squared Deviations

For each value \(N\), calculate \(N - m\) and \((N - m)^2\):

• For \(N = 10\): \(10 - 35=-25\), \((-25)^2 = 625\)
• For \(N = 15\): \(15 - 35=-20\), \((-20)^2 = 400\)
• For \(N = 25\): \(25 - 35=-10\), \((-10)^2 = 100\)
• For \(N = 35\): \(35 - 35 = 0\), \(0^2 = 0\)
• For \(N = 50\): \(50 - 35 = 15\), \(15^2 = 225\)
• For \(N = 75\): \(75 - 35 = 40\), \(40^2 = 1600\)

Step4: Calculate the Variance

Variance (for population) is the average of the squared deviations. The sum of squared deviations is \(625 + 400 + 100 + 0 + 225 + 1600 = 2950\). Variance \(\sigma^2=\frac{2950}{6}\approx491.67\) (if sample, divide by \(n - 1\), but here we assume population).

Step5: Calculate the Standard Deviation

Standard deviation is the square root of the variance. So, \(\sigma=\sqrt{\frac{2950}{6}}\approx\sqrt{491.67}\approx22.17\)

Answer:

• Mean: \(35\)
• Range: \(65\)
• Variance: \(\approx491.67\) (population) or \(\frac{2950}{5} = 590\) (sample)
• Standard Deviation: \(\approx22.17\) (population) or \(\sqrt{590}\approx24.29\) (sample)

(Note: If it's a sample, we use \(n - 1\) in variance calculation. The problem doesn't specify, so both are provided for reference.)