The weights (in pounds) of a sample of five boxes being sent by UPS are: 12, 6, 7, 3, and 10.
a. Compute the range.
b. Compute the mean deviation.
c. Compute the standard deviation.
http://www.purplemath.com/modules/meanmode.htm
http://www.google.com/search?rlz=1C1GGLS_en-USUS292&sourceid=chrome&ie=UTF-8&q=mean+deviation
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Let us know what you come up with.
Range=12-3=9
Mean=(12+6+7+3+10)/5=7.6
Mean Deviation=[(12-7.6)+(7.6-6)+(7.6-7)+(7.6-3)+(10-7.6)]/5 = 13.6/5=2.72
Standard Deviation=sqrt=[(12-7.6)^2(7.6-6)^2+(7.6-7)^2+(7.6-3)^2+(10-7.6)^2]/(5-1)=sqrt[19.36+2.56+0.36+21.16+ 5.76]/4=sqrt[12.3]=3.51
To compute the range, you need to find the difference between the highest and lowest values in the data set.
a. Compute the range:
The highest weight is 12, and the lowest weight is 3.
Range = Highest weight - Lowest weight
Range = 12 - 3
Range = 9 pounds
To compute the mean deviation, you need to find the average difference between each value and the mean of the data set.
b. Compute the mean deviation:
Step 1: Find the mean (average) of the weights:
Mean = (12 + 6 + 7 + 3 + 10) / 5
Mean = 38 / 5
Mean = 7.6 pounds
Step 2: Find the deviation for each weight:
Deviation = Weight - Mean
|Weight | Deviation |
|-------|-----------|
| 12 | 4.4 |
| 6 | -1.6 |
| 7 | -0.6 |
| 3 | -4.6 |
| 10 | 2.4 |
Step 3: Find the absolute deviation for each weight:
Absolute Deviation = |Deviation|
|Weight | Deviation | Absolute Deviation |
|-------|-----------|--------------------|
| 12 | 4.4 | 4.4 |
| 6 | -1.6 | 1.6 |
| 7 | -0.6 | 0.6 |
| 3 | -4.6 | 4.6 |
| 10 | 2.4 | 2.4 |
Step 4: Find the sum of the absolute deviations:
Sum of Absolute Deviations = 4.4 + 1.6 + 0.6 + 4.6 + 2.4
Sum of Absolute Deviations = 13.6 pounds
Step 5: Find the mean deviation:
Mean Deviation = Sum of Absolute Deviations / Number of data points
Mean Deviation = 13.6 / 5
Mean Deviation = 2.72 pounds
To compute the standard deviation, you can use the formula that uses the mean deviation.
c. Compute the standard deviation:
Standard Deviation = √(∑(Deviation^2) / Number of data points)
Step 1: Find the squared deviation for each weight:
Squared Deviation = Deviation^2
|Weight | Deviation | Squared Deviation |
|-------|-----------|-----------------------|
| 12 | 4.4 | 19.36 |
| 6 | -1.6 | 2.56 |
| 7 | -0.6 | 0.36 |
| 3 | -4.6 | 21.16 |
| 10 | 2.4 | 5.76 |
Step 2: Find the sum of the squared deviations:
Sum of Squared Deviations = 19.36 + 2.56 + 0.36 + 21.16 + 5.76
Sum of Squared Deviations = 49.2 pounds^2
Step 3: Find the average of the squared deviations:
Average of Squared Deviations = Sum of Squared Deviations / Number of data points
Average of Squared Deviations = 49.2 / 5
Average of Squared Deviations = 9.84 pounds^2
Step 4: Find the standard deviation:
Standard Deviation = √(Average of Squared Deviations)
Standard Deviation = √(9.84)
Standard Deviation ≈ 3.137 pounds
Therefore, the range is 9 pounds, the mean deviation is 2.72 pounds, and the standard deviation is approximately 3.137 pounds.
To compute the range, mean deviation, and standard deviation of the given set of weights, follow these steps:
a. Compute the range:
1. Find the smallest and largest weights in the given set.
Smallest weight: 3 pounds
Largest weight: 12 pounds
2. Subtract the smallest weight from the largest weight to obtain the range.
Range = Largest weight - Smallest weight
Range = 12 pounds - 3 pounds
Range = 9 pounds
Therefore, the range of the weights is 9 pounds.
b. Compute the mean deviation:
1. Calculate the mean (average) of the given weights.
Mean = (12 + 6 + 7 + 3 + 10) / 5
Mean = 38 / 5
Mean = 7.6 pounds
2. Subtract the mean from each weight and find the absolute value of the difference.
Absolute deviation = |Weight - Mean|
Absolute deviation: (12 - 7.6), (6 - 7.6), (7 - 7.6), (3 - 7.6), (10 - 7.6)
Absolute deviation: 4.4, 1.6, 0.6, 4.6, 2.4 pounds
3. Calculate the mean of these absolute deviations.
Mean deviation = (4.4 + 1.6 + 0.6 + 4.6 + 2.4) / 5
Mean deviation = 13.6 / 5
Mean deviation = 2.72 pounds
Therefore, the mean deviation of the weights is 2.72 pounds.
c. Compute the standard deviation:
1. Calculate the squared deviation of each weight from the mean.
Squared deviation = (Weight - Mean)^2
Squared deviation: (12 - 7.6)^2, (6 - 7.6)^2, (7 - 7.6)^2, (3 - 7.6)^2, (10 - 7.6)^2
Squared deviation: 19.36, 2.56, 0.36, 18.56, 5.76 square pounds
2. Find the mean of these squared deviations.
Mean squared deviation = (19.36 + 2.56 + 0.36 + 18.56 + 5.76) / 5
Mean squared deviation = 46.6 / 5
Mean squared deviation = 9.32 square pounds
3. Take the square root of the mean squared deviation to get the standard deviation.
Standard deviation = sqrt(Mean squared deviation)
Standard deviation = sqrt(9.32)
Standard deviation ≈ 3.05 pounds
Therefore, the standard deviation of the weights is approximately 3.05 pounds.