Correcting Observations in Statistical Analysis: A Practical Example

The importance of accurate data in statistical analysis cannot be overstated. When errors or inaccuracies are identified in data sets, it is crucial to correct them to ensure reliable results. This article will walk through a practical example where an observation is corrected, and the mean and standard deviation are recalculated.

The Initial Data Set and Observation Error

Consider a data set consisting of 20 observations with an initial mean of 31. It was later discovered that an observation was recorded as 52 instead of the correct value of 25. This error needs to be corrected to reflect the true mean of the data set.

Step 1: Correcting the Mean

The initial mean of the 20 observations is 31. To correct for the observation error, we need to adjust the sum of the observations. Calculate the total sum of the initial 20 observations: (20 times 31 620). Subtract the incorrect observation (52): (620 - 52 568). Add the correct observation (25): (568 25 593). Divide by the number of observations (19) to get the corrected mean: (593 div 19 31.211).

Thus, the corrected mean is approximately 31.211, more accurately calculated as 30.263.

Step 2: Correcting the Standard Deviation

The corrected standard deviation requires a stepwise adjustment. Let's break down the process:

Initial Calculation of the Sum of Squares (SS)

The sum of the deviations from the mean squared (SS) for the initial data set is given as 180. Subtract the deviation of the incorrect observation (52) from the mean (30), which is 22, squared: (22^2 484). Subtract this from the initial SS: (180 - 484 45^2 180 - 484 25^2 180 - 484 625 321).

Correcting the Sum of Squares (SS)

The corrected SS for the data set is 155. Divide by the number of observations minus one (19) to get the corrected variance: (frac{155}{19} 8.157). Take the square root to get the standard deviation: (sqrt{8.157} approx 2.856).

Conclusion

Through this example, we've demonstrated how to accurately correct a data set by identifying and then adjusting the mean and standard deviation based on observed data errors. This process ensures that the analysis reflects the true state of the data and provides reliable statistical insights.

Conceptually, the key principles are:

Initial mean: (31) Corrected sum: (593) Corrected mean: (30.263) Corrected sum of squares (SS): (155) Corrected variance: (8.157) Corrected standard deviation: (2.856)

For more detailed analyses or multiple observations, similar adjustments can be made following these steps.