Chapter Test
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Mean: 15
Median: 13.875
Mode: Does not exist.
Range: 15.70
Standard Deviation: Approximately 5.46 .
The first thing that should be done when finding the key features of a data set is rearranging the numbers from least to greatest. $7.80 $9.75 $10.60 $12.25 $15.50 [0.3em] $18.90 $21.70 $23.50 Let's proceed to finding the mean, median, mode, range, and standard deviation.
Substitute values
Add terms
Calculate quotient
We found that x=15.
To identify the median, we observe the middle value. 7.80 9.75 10.60 12.25 | 15.50 18.90 21.70 23.50 There is no middle value. When this happens, we need to calculate the median by finding the average of the two values closest to the middle. When arranged from least to greatest, 12.25 and 15.50 are the most central values. Median: 12.25+ 15.50/2=13.875
The mode of a data set is the value that occurs most frequently. 7.80 9.75 10.60 12.25 15.50 [0.3em] 18.90 21.70 23.50 In this set, each number occurs only once. Therefore, there is no mode.
The range is the difference between the greatest and least values in a set of data. For our exercise, the greatest value is 23.50 and the least value is 7.80. Range: 23.50- 7.80= 15.70
The standard deviation of a set of data is the average amount by which each individual value deviates or differs from the mean. Standard Deviation sqrt((x_1-x )^2 + (x_2-x )^2 + ... + (x_n-x )^2/n) In the above formula, x_1, ... ,x_n are the values of the set of data, x is the mean, and n is the number of values. For this exercise we have 8 values and we already found the mean to be equal to 15. Let's use this value and apply the formula to each value in the set.
| x_n | x_n-x | (x_n-x)^2 |
|---|---|---|
| 7.80 | 7.80-15=- 7.20 | (- 7.20)^2 = 51.84 |
| 9.75 | 9.75-15=- 5.25 | (- 5.25)^2 ≈ 27.56 |
| 10.60 | 10.60-15=- 4.40 | (- 4.40)^2 = 19.36 |
| 12.25 | 12.25-15=- 2.75 | (- 2.75)^2 ≈ 7.56 |
| 15.50 | 15.50-15=0.50 | 0.50^2 = 0.25 |
| 18.90 | 18.90-15=3.90 | 3.90^2 = 15.21 |
| 21.70 | 21.70-15=6.70 | 6.70^2 = 44.89 |
| 23.50 | 23.50-15=8.5 | 8.5^2 = 72.25 |
| Sum of Values | ≈ 238.92 | |
Finally, since n= 8, we need to divide by 8 and then calculate the square root. Standard Deviation: sqrt(238.92/8)≈ 5.46