Sure! Let's go through the problem step-by-step.
First, we will address part (a), calculating the range of the given data set. The range is found by taking the difference between the maximum and minimum values in the data set.
The data set is: 48, 50, 45, 55, 46, 54, 52.
Step 1: Identify the maximum value and the minimum value in the data set.
- Maximum value: 55
- Minimum value: 45
Step 2: Compute the range by subtracting the minimum value from the maximum value.
- Range = Maximum value - Minimum value
- Range = 55 - 45
So, the range is 10.
Now moving on to part (b), calculating the standard deviation. The standard deviation is a measure of the amount of variation or dispersion in a set of values.
Step 1: Find the mean (average) of the data set.
[tex]\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \][/tex]
[tex]\[ \text{Mean} = \frac{48 + 50 + 45 + 55 + 46 + 54 + 52}{7} \][/tex]
[tex]\[ \text{Mean} = \frac{350}{7} \][/tex]
[tex]\[ \text{Mean} = 50 \][/tex]
Step 2: Calculate the variance. Variance is found by taking the average of the squared differences from the mean.
[tex]\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \][/tex]
Where [tex]\( x_i \)[/tex] represents each data point and [tex]\( N \)[/tex] is the number of data points.
Let's compute each squared difference from the mean:
- (48 - 50)^2 = 4
- (50 - 50)^2 = 0
- (45 - 50)^2 = 25
- (55 - 50)^2 = 25
- (46 - 50)^2 = 16
- (54 - 50)^2 = 16
- (52 - 50)^2 = 4
Sum of these squared differences:
[tex]\[ 4 + 0 + 25 + 25 + 16 + 16 + 4 = 90 \][/tex]
Step 3: Find the variance by dividing this sum by the number of data points.
[tex]\[ \text{Variance} = \frac{90}{7} \approx 12.857 \][/tex]
Step 4: The standard deviation is the square root of the variance.
[tex]\[ \text{Standard Deviation} \approx \sqrt{12.857} \approx 3.586 \][/tex]
So, the answers are:
(a) The range is 10.
(b) The standard deviation is 3.586 (rounded to the nearest thousandth).
