The mean height of 30 students was given as 140 cm. It was later observed that one of the observations was incorrectly copied as 121 cm while it was actually 134 cm. What was the correct mean height? [Give your answer correct to one place of decimal.]

Question

Prepp Team · Accepted Answer

Calculating the Correct Mean Height of Students

The mean is a measure of central tendency, calculated by dividing the sum of all observations by the total number of observations. In this problem, we are given the mean height of 30 students and information about an incorrect observation that needs correction to find the accurate mean height.

Understanding the Initial Data

We are given the following initial information:

Number of students (n) = 30
Initial mean height = 140 cm

Using the formula for the mean:

$$\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}$$

We can find the initial sum of the heights of the 30 students:

$$\text{Initial Sum of Heights} = \text{Initial Mean} \times \text{Number of Students}$$

$$\text{Initial Sum of Heights} = 140 \text{ cm} \times 30 = 4200 \text{ cm}$$

Correcting the Observation Error

It was found that one observation was incorrectly copied. We have:

Incorrectly recorded height = 121 cm
Correct height = 134 cm

To find the correct sum of heights, we need to subtract the incorrect value from the initial sum and add the correct value:

$$\text{Correct Sum of Heights} = \text{Initial Sum of Heights} - \text{Incorrect Height} + \text{Correct Height}$$

$$\text{Correct Sum of Heights} = 4200 \text{ cm} - 121 \text{ cm} + 134 \text{ cm}$$

$$\text{Correct Sum of Heights} = 4200 \text{ cm} + (134 - 121) \text{ cm}$$

$$\text{Correct Sum of Heights} = 4200 \text{ cm} + 13 \text{ cm}$$

$$\text{Correct Sum of Heights} = 4213 \text{ cm}$$

Calculating the Correct Mean Height

Now that we have the correct sum of heights and the number of students (which remains 30), we can calculate the correct mean height:

$$\text{Correct Mean Height} = \frac{\text{Correct Sum of Heights}}{\text{Number of Students}}$$

$$\text{Correct Mean Height} = \frac{4213 \text{ cm}}{30}$$

Let's perform the division:

$$4213 \div 30$$

$$4213 / 30 \approx 140.4333...$$

We need to give the answer correct to one place of decimal. Rounding 140.4333... to one decimal place gives 140.4 cm.

Calculation Step	Value
Initial Number of Students	30
Initial Mean Height	140 cm
Initial Sum of Heights	$140 \times 30 = 4200$ cm
Incorrect Observation	121 cm
Correct Observation	134 cm
Correct Sum of Heights	$4200 - 121 + 134 = 4213$ cm
Correct Mean Height	$4213 / 30 \approx 140.43$ cm
Correct Mean (1 decimal place)	140.4 cm

Therefore, the correct mean height of the 30 students is 140.4 cm.

Revision Table: Key Concepts in Mean Calculation

Concept	Description	Formula
Mean (Arithmetic Mean)	The average of a set of numbers.	$\frac{\text{Sum of values}}{\text{Number of values}}$
Sum of Observations	The total obtained by adding all the observations in a dataset.	Mean $\times$ Number of Observations
Correcting Mean	Adjusting the mean when an observation was recorded incorrectly.	$\frac{\text{Original Sum} - \text{Incorrect Value} + \text{Correct Value}}{\text{Total Number of Observations}}$

Additional Information: Importance of Accurate Data

Accurate data is crucial for calculating reliable statistics like the mean. An error in even a single observation can affect the sum and consequently the mean. Problems like this one highlight the importance of data cleaning and verification before performing statistical analysis. Ensuring data accuracy leads to more meaningful and trustworthy conclusions drawn from the data.

When dealing with large datasets, identifying and correcting errors can be more complex, often requiring statistical techniques or data validation procedures. However, the basic principle remains the same: subtract the incorrect value and add the correct value to the total sum before calculating the corrected mean.