An article in archaeometry involved an analysis of 26 samples of Romano-British pottery found at four different kiln sites in the United Kingdom. The samples were analyzed to determine their composition. The percentage of iron oxide in each of five samples collected at the island thorns site was : 1.28, 2.39, 1.50, 1.88, 1.51. Calculate the sample variance and the standard deviation using the computing formula.
To calculate the sample variance and standard deviation, we need to follow these steps:
Step 1: Calculate the mean (average) of the data set.
Step 2: Subtract the mean from each data point.
Step 3: Square each result from step 2.
Step 4: Sum up all the squared values from step 3.
Step 5: Divide the sum from step 4 by (n - 1), where n is the number of data points. This gives you the sample variance.
Step 6: Take the square root of the sample variance to calculate the sample standard deviation.
Now let's calculate step by step:
Step 1: Calculate the mean (average)
(mean) x̄ = (1.28 + 2.39 + 1.50 + 1.88 + 1.51) / 5
(mean) x̄ = 8.56 / 5
(mean) x̄ = 1.712
Step 2: Subtract the mean from each data point.
1.28 - 1.712 = -0.432
2.39 - 1.712 = 0.678
1.50 - 1.712 = -0.212
1.88 - 1.712 = 0.168
1.51 - 1.712 = -0.202
Step 3: Square each result from step 2.
(-0.432)^2 = 0.186624
(0.678)^2 = 0.459684
(-0.212)^2 = 0.044944
(0.168)^2 = 0.028224
(-0.202)^2 = 0.040804
Step 4: Sum up all the squared values from step 3.
0.186624 + 0.459684 + 0.044944 + 0.028224 + 0.040804 = 0.760280
Step 5: Divide the sum from step 4 by (n - 1) to get the sample variance.
(sample variance) S^2 = 0.760280 / (5 - 1)
(sample variance) S^2 = 0.760280 / 4
(sample variance) S^2 = 0.190070
Step 6: Take the square root of the sample variance to get the sample standard deviation.
(sample standard deviation) S = √0.190070
(sample standard deviation) S ≈ 0.435961
Therefore, the sample variance is 0.190070 and the sample standard deviation is approximately 0.435961.
To calculate the sample variance and standard deviation using the computing formula, follow these steps:
Step 1: Calculate the mean (average) of the data set.
To find the mean, add up all the values and divide by the number of values (in this case, 5):
Mean (µ) = (1.28 + 2.39 + 1.50 + 1.88 + 1.51) / 5
Mean (µ) = 1.71
Step 2: Calculate the difference between each value and the mean.
Subtract the mean from each value and square the result:
(1.28 - 1.71)^2 = 0.1849
(2.39 - 1.71)^2 = 0.4624
(1.50 - 1.71)^2 = 0.0441
(1.88 - 1.71)^2 = 0.0289
(1.51 - 1.71)^2 = 0.0400
Step 3: Sum up all the squared differences.
Add up all the squared differences calculated in Step 2:
0.1849 + 0.4624 + 0.0441 + 0.0289 + 0.0400 = 0.7603
Step 4: Calculate the sample variance.
Divide the sum of squared differences (from Step 3) by the number of values minus 1 (4 in this case):
Sample Variance (s^2) = 0.7603 / (5 - 1)
Sample Variance (s^2) = 0.7603 / 4
Sample Variance (s^2) = 0.1901
Step 5: Calculate the standard deviation.
Take the square root of the sample variance to find the standard deviation:
Standard Deviation (s) = √(0.1901)
Standard Deviation (s) ≈ 0.4366
Therefore, the sample variance is approximately 0.1901 and the standard deviation is approximately 0.4366.