In successive experimental measurements, the refractive index of a glass turned out to be 1.54, 1.45, 1.53, 1.56, 1.44, 1.54. Calculate:
i. Mean refractive index
ii. Mean absolute error
iii. Fractional error
iv. Percentage error
To calculate the mean refractive index, mean absolute error, fractional error, and percentage error, follow these steps:
Step 1: Calculate the mean refractive index.
To find the mean refractive index, add up all the values and divide by the total number of measurements.
Mean Refractive Index = (1.54 + 1.45 + 1.53 + 1.56 + 1.44 + 1.54) / 6
Mean Refractive Index = 9.06 / 6
Mean Refractive Index = 1.51
Step 2: Calculate the mean absolute error.
To find the mean absolute error, subtract each measurement from the mean refractive index, take the absolute value of each difference, and then compute the average of these absolute differences.
Mean Absolute Error = |1.54 - 1.51| + |1.45 - 1.51| + |1.53 - 1.51| + |1.56 - 1.51| + |1.44 - 1.51| + |1.54 - 1.51| / 6
Mean Absolute Error = 0.03 + 0.06 + 0.02 + 0.05 + 0.07 + 0.03 / 6
Mean Absolute Error = 0.26 / 6
Mean Absolute Error = 0.0433
Step 3: Calculate the fractional error.
To find the fractional error, divide the mean absolute error by the mean refractive index.
Fractional Error = Mean Absolute Error / Mean Refractive Index
Fractional Error = 0.0433 / 1.51
Fractional Error ≈ 0.0287
Step 4: Calculate the percentage error.
To find the percentage error, multiply the fractional error by 100.
Percentage Error = Fractional Error * 100
Percentage Error ≈ 0.0287 * 100
Percentage Error ≈ 2.87%
Therefore:
i. Mean refractive index = 1.51
ii. Mean absolute error = 0.0433
iii. Fractional error ≈ 0.0287
iv. Percentage error ≈ 2.87%
To calculate the required values, follow these steps:
i. Mean refractive index:
1. Add up all the values: 1.54 + 1.45 + 1.53 + 1.56 + 1.44 + 1.54 = 9.06.
2. Divide the sum by the number of measurements: 9.06 / 6 = 1.51.
Therefore, the mean refractive index is 1.51.
ii. Mean absolute error:
1. Subtract the mean refractive index from each measurement:
|1.54 - 1.51| = 0.03, |1.45 - 1.51| = 0.06, |1.53 - 1.51| = 0.02,
|1.56 - 1.51| = 0.05, |1.44 - 1.51| = 0.07, |1.54 - 1.51| = 0.03.
2. Add up all the absolute errors: 0.03 + 0.06 + 0.02 + 0.05 + 0.07 + 0.03 = 0.26.
3. Divide the sum by the number of measurements: 0.26 / 6 = 0.0433 (rounded to four decimal places).
Therefore, the mean absolute error is 0.0433.
iii. Fractional error:
1. Divide the mean absolute error by the mean refractive index:
0.0433 / 1.51 ≈ 0.0287 (rounded to four decimal places).
Therefore, the fractional error is approximately 0.0287.
iv. Percentage error:
1. Multiply the fractional error by 100 to obtain the percentage:
0.0287 * 100 ≈ 2.87% (rounded to two decimal places).
Therefore, the percentage error is approximately 2.87%.