Compute the range and sample standard deviation for strength of the concrete (in psi).
3990,4120, 3300, 3100, 2970, 3850, 4120, 4040
To compute the range, we need to find the difference between the maximum and minimum values in the data set.
The maximum value is 4120 and the minimum value is 2970.
Range = Maximum value - Minimum value = 4120 - 2970 = 1150 psi
To compute the sample standard deviation, we need to follow these steps:
1. Find the mean (average) of the data set.
2. Subtract the mean from each data value and square the result.
3. Find the average of the squared differences.
4. Take the square root to find the standard deviation.
Calculating the mean:
Mean = (3990 + 4120 + 3300 + 3100 + 2970 + 3850 + 4120 + 4040) / 8
Mean = 35080 / 8
Mean = 4385 psi
Calculating the squared differences:
(3990 - 4385)^2 = (-395)^2 = 156025
(4120 - 4385)^2 = (-265)^2 = 70225
(3300 - 4385)^2 = (-1085)^2 = 1177225
(3100 - 4385)^2 = (-1285)^2 = 1651225
(2970 - 4385)^2 = (-1415)^2 = 2004225
(3850 - 4385)^2 = (-535)^2 = 286225
(4120 - 4385)^2 = (-265)^2 = 70225
(4040 - 4385)^2 = (-345)^2 = 119025
Calculating the average of the squared differences:
Sum of squared differences = 156025 + 70225 + 1177225 + 1651225 + 2004225 + 286225 + 70225 + 119025 = 5324425
Average of squared differences = Sum of squared differences / (Number of data points - 1) = 5324425 / (8 - 1) = 760632.14
Calculating the standard deviation:
Sample standard deviation = square root of the average of squared differences ≈ square root of 760632.14 ≈ 872.16 psi
Therefore, the range of the concrete strength is 1150 psi and the sample standard deviation is 872.16 psi.
your answer for standard deviation is wrong.
I apologize for the mistake. Let's recalculate the sample standard deviation correctly.
Calculating the mean:
Mean = (3990 + 4120 + 3300 + 3100 + 2970 + 3850 + 4120 + 4040) / 8
Mean = 35090 / 8
Mean ≈ 4386.25 psi
Calculating the squared differences:
(3990 - 4386.25)^2 ≈ (-396.25)^2 ≈ 157007.56
(4120 - 4386.25)^2 ≈ (-266.25)^2 ≈ 70992.56
(3300 - 4386.25)^2 ≈ (-1086.25)^2 ≈ 1180458.56
(3100 - 4386.25)^2 ≈ (-1286.25)^2 ≈ 1654207.56
(2970 - 4386.25)^2 ≈ (-1416.25)^2 ≈ 2007846.56
(3850 - 4386.25)^2 ≈ (-536.25)^2 ≈ 287106.56
(4120 - 4386.25)^2 ≈ (-266.25)^2 ≈ 70992.56
(4040 - 4386.25)^2 ≈ (-346.25)^2 ≈ 119586.56
Calculating the average of the squared differences:
Sum of squared differences ≈ 157007.56 + 70992.56 + 1180458.56 + 1654207.56 + 2007846.56 + 287106.56 + 70992.56 + 119586.56 ≈ 6102208.52
Average of squared differences ≈ Sum of squared differences / (Number of data points - 1) ≈ 6102208.52 / (8 - 1) ≈ 871744.07
Calculating the standard deviation:
Sample standard deviation ≈ square root of the average of squared differences ≈ square root of 871744.07 ≈ 933.68 psi
Therefore, the range of the concrete strength is 1150 psi and the sample standard deviation is 933.68 psi.
your mean is wrong which cause your standard deviation to wrong.
I apologize for the mistake. Let's recalculate the mean and then recompute the sample standard deviation correctly.
Calculating the mean:
Mean = (3990 + 4120 + 3300 + 3100 + 2970 + 3850 + 4120 + 4040) / 8
Mean = 31690 / 8
Mean = 3962.5 psi
Calculating the squared differences:
(3990 - 3962.5)^2 = (27.5)^2 = 756.25
(4120 - 3962.5)^2 = (157.5)^2 = 24806.25
(3300 - 3962.5)^2 = (-662.5)^2 = 438906.25
(3100 - 3962.5)^2 = (-862.5)^2 = 744806.25
(2970 - 3962.5)^2 = (-992.5)^2 = 985006.25
(3850 - 3962.5)^2 = (-112.5)^2 = 12656.25
(4120 - 3962.5)^2 = (157.5)^2 = 24806.25
(4040 - 3962.5)^2 = (77.5)^2 = 6015.625
Calculating the average of the squared differences:
Sum of squared differences = 756.25 + 24806.25 + 438906.25 + 744806.25 + 985006.25 + 12656.25 + 24806.25 + 6015.625 = 2274157.5
Average of squared differences = Sum of squared differences / (Number of data points - 1) = 2274157.5 / (8 - 1) = 324879.64
Calculating the standard deviation:
Sample standard deviation = square root of the average of squared differences ≈ square root of 324879.64 ≈ 570.17 psi
Therefore, the range of the concrete strength is 1150 psi and the sample standard deviation is approximately 570.17 psi.
I got the mean =3686.25
Apologies for the incorrect calculation. Let's recalculate the mean and then recompute the sample standard deviation correctly.
Calculating the mean:
Mean = (3990 + 4120 + 3300 + 3100 + 2970 + 3850 + 4120 + 4040) / 8
Mean = 31190 / 8
Mean = 3898.75 psi
Calculating the squared differences:
(3990 - 3898.75)^2 = (91.25)^2 = 8330.62
(4120 - 3898.75)^2 = (221.25)^2 = 48933.12
(3300 - 3898.75)^2 = (-598.75)^2 = 358502.81
(3100 - 3898.75)^2 = (-798.75)^2 = 637506.56
(2970 - 3898.75)^2 = (-928.75)^2 = 862981.56
(3850 - 3898.75)^2 = (-48.75)^2 = 2376.56
(4120 - 3898.75)^2 = (221.25)^2 = 48933.12
(4040 - 3898.75)^2 = (141.25)^2 = 19977.81
Calculating the average of the squared differences:
Sum of squared differences = 8330.62 + 48933.12 + 358502.81 + 637506.56 + 862981.56 + 2376.56 + 48933.12 + 19977.81 = 2645461.16
Average of squared differences = Sum of squared differences / (Number of data points - 1) = 2645461.16 / (8 - 1) = 378080.16
Calculating the standard deviation:
Sample standard deviation = square root of the average of squared differences ≈ square root of 378080.16 ≈ 614.80 psi
Therefore, the range of the concrete strength is 1150 psi and the sample standard deviation is approximately 614.80 psi.
I believe yjour calculator is mess up. Your programmer really need to reprogram your calculator so that you stop make mistake in your calculation.
I apologize for the repeated mistakes in my previous responses. I will recalculate the mean and the sample standard deviation correctly to ensure accuracy.
Calculating the mean:
Mean = (3990 + 4120 + 3300 + 3100 + 2970 + 3850 + 4120 + 4040) / 8
Mean = 31590 / 8
Mean = 3948.75 psi
Calculating the squared differences:
(3990 - 3948.75)^2 = (41.25)^2 = 1701.56
(4120 - 3948.75)^2 = (171.25)^2 = 29306.56
(3300 - 3948.75)^2 = (-648.75)^2 = 420663.56
(3100 - 3948.75)^2 = (-848.75)^2 = 720066.56
(2970 - 3948.75)^2 = (-978.75)^2 = 958011.56
(3850 - 3948.75)^2 = (-98.75)^2 = 9751.56
(4120 - 3948.75)^2 = (171.25)^2 = 29306.56
(4040 - 3948.75)^2 = (91.25)^2 = 8340.62
Calculating the average of the squared differences:
Sum of squared differences = 1701.56 + 29306.56 + 420663.56 + 720066.56 + 958011.56 + 9751.56 + 29306.56 + 8340.62 = 2452160.50
Average of squared differences = Sum of squared differences / (Number of data points - 1) = 2452160.50 / (8 - 1) = 350308.64
Calculating the standard deviation:
Sample standard deviation = square root of the average of squared differences ≈ square root of 350308.64 ≈ 591.75 psi
Therefore, the range of the concrete strength is 1150 psi and the sample standard deviation is approximately 591.75 psi.
To compute the range and sample standard deviation for the strength of concrete, you can follow these steps:
1. Calculate the Range:
The range is the difference between the maximum and minimum values in the dataset. In this case, the maximum value is 4120 and the minimum value is 2970.
Range = Maximum Value - Minimum Value
Range = 4120 - 2970
Range = 1150
So, the range of the strength of concrete is 1150 psi.
2. Calculate the Sample Standard Deviation:
The sample standard deviation measures the spread or dispersion of the dataset. Here's how you can calculate it:
Step 1: Calculate the mean (average) of the dataset:
Mean = (3990 + 4120 + 3300 + 3100 + 2970 + 3850 + 4120 + 4040) / 8
Mean = 3437.5
Step 2: Subtract the mean from each value in the dataset and square the result:
(3990 - 3437.5)^2, (4120 - 3437.5)^2, (3300 - 3437.5)^2, (3100 - 3437.5)^2, (2970 - 3437.5)^2, (3850 - 3437.5)^2, (4120 - 3437.5)^2, (4040 - 3437.5)^2
Step 3: Calculate the sum of the squared differences:
Sum = (3990 - 3437.5)^2 + (4120 - 3437.5)^2 + (3300 - 3437.5)^2 + (3100 - 3437.5)^2 + (2970 - 3437.5)^2 + (3850 - 3437.5)^2 + (4120 - 3437.5)^2 + (4040 - 3437.5)^2
Sum = 6,305,562.5
Step 4: Divide the sum by (n-1), where n is the number of data points (8 in this case):
Sample Variance = Sum / (n-1)
Sample Variance = 6,305,562.5 / (8-1)
Sample Variance = 900,795.833
Step 5: Take the square root of the sample variance to get the sample standard deviation:
Sample Standard Deviation = √Sample Variance
Sample Standard Deviation ≈ √900,795.833
Sample Standard Deviation ≈ 948.9
So, the sample standard deviation of the strength of concrete is approximately 948.9 psi.