Question

For the datasets. Use technology to find the following values: (a) The mean and the standard deviation. (b) The five number summary. 25, 72, 77, 31, 80, 80, 64, 39, 75, 58, 43, 67, 54, 71, 60

Short answer

The mean is around 60.53, the standard deviation is about 18.69. The five-number summary is: min=25, Q1=approx. 43.5, median=64, Q3=approx.75.5, max=80.

Step-by-step solution

Calculate the Mean

To calculate the mean, add all the numbers together and divide by the number of numbers. Here, there are 15 numbers, so the mean is \((25+72+77+31+80+80+64+39+75+58+43+67+54+71+60) / 15\).

Calculate the Standard Deviation

The standard deviation measures the variance in a set of data. To calculate it, follow these steps: 1. Find the mean.2. Subtract the mean from each number and square the result. 3. Find the mean of these squared differences. 4. Take the square root of that number. Use this formula: \(\sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \mu)^2}\). In this formula, \(x_i\) represents each value in the dataset, \(\mu\) is the mean, \(n\) is the number of values, and \(\sum\) is the sum.

Calculate the Five Number Summary

The five number summary includes the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. Min = 25, Max = 80. To find Q1, arrange the dataset in increasing order. The value which divides the first 25% data from the rest is Q1. To find Q3, look for the value that divides first 75% data from the rest. And the median is the value in the middle when the dataset is ordered from least to greatest.