Question

1. you are training for a triathlon. the miles that you ride your bicycle for 7 weeks are given as follows: 17, 20, 16, 18, 22, 19, 20 find the mean, median, mode, range, standard deviation of the data set.

Explanation:

Step1: Calculate the Mean

The mean is the sum of all values divided by the number of values. First, sum the data: \(17 + 20 + 16 + 18 + 22 + 19 + 20 = 132\). There are 7 values, so the mean is \(\frac{132}{7} \approx 18.86\) (rounded to two decimal places).

Step2: Find the Median

First, order the data set: \(16, 17, 18, 19, 20, 20, 22\). The median is the middle value. Since there are 7 values (odd number), the middle one is the 4th value, which is \(19\).

Step3: Determine the Mode

The mode is the value that appears most frequently. In the data set, \(20\) appears twice, and all other values appear once, so the mode is \(20\).

Step4: Calculate the Range

The range is the difference between the maximum and minimum values. The maximum value is \(22\) and the minimum is \(16\), so the range is \(22 - 16 = 6\).

Step5: Compute the Standard Deviation

First, find the variance. The formula for variance (\(\sigma^2\)) of a sample (if it were a sample) or population (since we have all 7 weeks, we can consider it a population) is \(\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}\), where \(\mu\) is the mean and \(N\) is the number of values.

1. Calculate each \((x_i - \mu)^2\):

• For \(16\): \((16 - 18.86)^2 \approx (-2.86)^2 \approx 8.18\)
• For \(17\): \((17 - 18.86)^2 \approx (-1.86)^2 \approx 3.46\)
• For \(18\): \((18 - 18.86)^2 \approx (-0.86)^2 \approx 0.74\)
• For \(19\): \((19 - 18.86)^2 \approx (0.14)^2 \approx 0.02\)
• For \(20\): \((20 - 18.86)^2 \approx (1.14)^2 \approx 1.30\) (two values of 20, so \(2 \times 1.30 = 2.60\))
• For \(22\): \((22 - 18.86)^2 \approx (3.14)^2 \approx 9.86\)

1. Sum these squared differences: \(8.18 + 3.46 + 0.74 + 0.02 + 2.60 + 9.86 = 24.86\)

1. Divide by \(N = 7\) to get the variance: \(\sigma^2 = \frac{24.86}{7} \approx 3.55\)

1. The standard deviation (\(\sigma\)) is the square root of the variance: \(\sigma = \sqrt{3.55} \approx 1.88\) (rounded to two decimal places)

Answer:

• Mean: \(\approx 18.86\)
• Median: \(19\)
• Mode: \(20\)
• Range: \(6\)
• Standard Deviation: \(\approx 1.88\)