Problem 12 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean of a data set consisting of 20 observations is 40 . If one observation 53 was wrongly recorded as 33 , then the correct mean will be : \(\quad\) [Online April9, 2013] (a) 41 (b) 49 (c) \(40.5\) (d) \(42.5\)

Step-by-Step Solution

Answer

The correct mean is 41.

Step 1: Identify the given information

The initial mean of the data set consisting of 20 observations is 40. This means the total sum of the data set is \(20 \times 40 = 800\). One observation (53) was recorded wrongly as 33.

Step 2: Calculate the correct total sum

Since 53 was recorded as 33, we need to adjust the total sum. Subtract the wrong observation and add the correct one: \(800 - 33 + 53 = 820\).

Step 3: Calculate the correct mean

Now, divide the corrected total sum by the number of observations to find the correct mean: \(\frac{820}{20} = 41\).

Key Concepts

Data Correction

In the realm of statistics, accurate data is crucial for precise calculations, as any incorrectly recorded data can lead to skewed results. Imagine you mistakenly record a number, but instead of ignoring it, you employ data correction to refine your dataset.
Data correction involves identifying and rectifying these inaccuracies. In our example, an observation was wrongly documented as 33, but its true value was 53. By recognizing this error, we adjusted the dataset to ensure our calculations were based on accurate data.
This process is vital because errors, if left uncorrected, can significantly affect measures such as the mean, variance, and other statistical calculations. Paying attention to detail and correcting errors ensures that your statistical analysis remains valid and reliable.

Observations in Data Set

Each individual value in a dataset is termed an observation. Observations are the building blocks of statistical analysis, collectively defining the dataset's characteristics. For instance, the data set in question has 20 observations.
These observations can represent a wide range of data types, such as measurements, survey responses, or recorded events. In our exercise, each observation contributes to calculating the mean, which reflects the dataset's central tendency.
When managing a dataset, it's essential to carefully review each observation for accuracy, as seen in the corrective steps taken for this particular problem. This meticulous review process reinforces the reliability of any statistical analysis performed using these observations.

Arithmetic Mean

The arithmetic mean, often referred to simply as the mean, is a statistic that represents the average of a data set. It's a fundamental measure of central tendency, providing insights into the general level of the observations.
Here's how you calculate the arithmetic mean: add up all the observations, then divide by the total number of observations. For example, when the total sum of a corrected dataset is noted as 820 and there are 20 observations, the mean is calculated as \( \frac{820}{20} = 41 \).
The arithmetic mean is widely used because it is straightforward and provides a good snapshot of the data set. However, it can be sensitive to outliers or incorrect values, so ensuring accurate data is key to meaningful statistical conclusions.