The mean and the standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who mistook one observation as 50 instead of 40. What are the correct mean and standard deviation?

As a seasoned tutor registered on UrbanPro, I'm here to guide you through this statistical hiccup. Let's address this query with precision.

Firstly, let me commend your commitment to mastering statistical concepts. Now, let's dive into the solution.

The mean (xˉxˉ) of a set of observations is the sum of all values divided by the number of observations. The standard deviation (σσ) is a measure of the dispersion or spread of a set of values from its mean.

Given:

• Original mean (xˉxˉ) = 40
• Original standard deviation (σσ) = 5.1
• Mistaken observation = 50

To correct the mean, we need to remove the mistaken observation and replace it with the correct value (40). Since there are 100 observations in total, the contribution of the mistaken observation to the mean is 50100=0.510050=0.5. So, we subtract 0.5 from the original mean and add the correct value (40):

Corrected mean (xˉcorrectedxˉcorrected) = xˉ−0.5+40xˉ−0.5+40

To correct the standard deviation, we need to recalculate it based on the corrected mean and observations. However, the standard deviation formula involves the square of the differences between each observation and the mean, so correcting one observation affects all the others. We must consider this correction while recalculating the standard deviation.

Therefore, it's not straightforward to calculate the corrected standard deviation manually. Instead, we can use statistical software or calculators to find the corrected standard deviation.

In summary, to find the correct mean, we subtract the contribution of the mistaken observation and add the correct value. To find the corrected standard deviation, we need to recalculate it using statistical tools due to the interconnected nature of the standard deviation formula.

If you need further assistance or guidance on statistical concepts or any other subject, feel free to reach out. Remember, UrbanPro is your ally in your academic journey, providing the best online coaching tuition.