What is the standard deviation of the data set given the atmospheric pressure readings in millibars: 1013, 1008, 1016, 1010, 1006, 1014, 1013, and 1009?
A. 4.583
B. 4.773
C. 3.446
D. 3.580
To calculate the standard deviation of a data set, you need to follow these steps:
1. Find the mean of the data set. The mean is calculated by summing up all the values in the data set and dividing it by the total number of values.
- In this case, the sum of the values is 1013 + 1008 + 1016 + 1010 + 1006 + 1014 + 1013 + 1009 = 8089. The total number of values is 8.
- So, the mean is 8089 / 8 = 1011.125.
2. Subtract the mean from each value in the data set, and square the result.
- Subtracting the mean from each value yields: 1013 - 1011.125 = 1.875, 1008 - 1011.125 = -3.125, 1016 - 1011.125 = 4.875, 1010 - 1011.125 = -1.125, 1006 - 1011.125 = -5.125, 1014 - 1011.125 = 2.875, 1013 - 1011.125 = 1.875, and 1009 - 1011.125 = -2.125.
- Squaring these differences gives: 1.875^2 = 3.515625, (-3.125)^2 = 9.765625, 4.875^2 = 23.765625, (-1.125)^2 = 1.265625, (-5.125)^2 = 26.265625, 2.875^2 = 8.265625, 1.875^2 = 3.515625, and (-2.125)^2 = 4.515625.
3. Find the sum of all the squared differences.
- The sum of the squared differences is 3.515625 + 9.765625 + 23.765625 + 1.265625 + 26.265625 + 8.265625 + 3.515625 + 4.515625 = 80.875.
4. Divide the sum of squared differences by the total number of values in the data set, which is n.
- In this case, n = 8, so the sum of squared differences divided by 8 is 80.875 / 8 = 10.109375.
5. Take the square root of the result from step 4 to find the standard deviation.
- The square root of 10.109375 is approximately 3.180678808.
Comparing the calculated standard deviation to the given options, none match exactly. However, the closest option is D. 3.580.
So, the answer is D. 3.580, which is the closest match to the calculated standard deviation of the data set.
To calculate the standard deviation of a data set, you can follow these steps:
Step 1: Find 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: Find the mean of the squared values.
Step 5: Take the square root of the mean from Step 4.
Now, let's calculate the standard deviation for the given data set: 1013, 1008, 1016, 1010, 1006, 1014, 1013, and 1009.
Step 1: Calculate the mean.
Mean = (1013 + 1008 + 1016 + 1010 + 1006 + 1014 + 1013 + 1009) / 8 = 1010.375
Step 2: Subtract the mean from each data point.
1013 - 1010.375 = 2.625
1008 - 1010.375 = -2.375
1016 - 1010.375 = 5.625
1010 - 1010.375 = -0.375
1006 - 1010.375 = -4.375
1014 - 1010.375 = 3.625
1013 - 1010.375 = 2.625
1009 - 1010.375 = -1.375
Step 3: Square each result from Step 2.
2.625^2 = 6.890625
(-2.375)^2 = 5.640625
5.625^2 = 31.765625
(-0.375)^2 = 0.140625
(-4.375)^2 = 19.140625
3.625^2 = 13.140625
2.625^2 = 6.890625
(-1.375)^2 = 1.890625
Step 4: Find the mean of the squared values.
Mean = (6.890625 + 5.640625 + 31.765625 + 0.140625 + 19.140625 + 13.140625 + 6.890625 + 1.890625) / 8 = 9.605469
Step 5: Take the square root of the mean from Step 4.
Standard Deviation = √9.605469 = 3.100563
Based on the calculations, the standard deviation of the given data set is approximately 3.100563, which is closest to option C. 3.446.