Understanding Variance Calculation for Largest Observations

This problem requires calculating the variance of a specific subset of data – the largest five observations from a list of student marks in a Statistics test. Variance measures how spread out the data points are from their average value (mean).

Step 1: Identify the Data Set

The marks obtained by 10 students are:

24, 47, 18, 32, 19, 15, 21, 35, 50, 41

Step 2: Find the Largest Five Observations

First, we need to sort the marks in ascending order to easily identify the largest values:

15, 18, 19, 21, 24, 32, 35, 41, 47, 50

The largest five observations are:

32, 35, 41, 47, 50

Step 3: Calculate the Mean of the Largest Five Observations

To find the variance, we first need the mean (\(\bar{x}\)) of these five numbers.

Mean (\(\bar{x}\)) = Sum of observations / Number of observations

\(\bar{x} = \frac{32 + 35 + 41 + 47 + 50}{5}\)

\(\bar{x} = \frac{205}{5}\)

\(\bar{x} = 41\)

Step 4: Calculate the Variance

Variance measures the average squared difference of each data point from the mean. Since we are considering these five observations as the complete set of interest for this calculation, we use the formula for population variance (\(\sigma^2\)):

Population Variance (\(\sigma^2\)) = \(\frac{\sum_{i=1}^{N}(x_i - \bar{x})^2}{N}\)

Where:

• \(x_i\) represents each of the five observations (32, 35, 41, 47, 50).
• \(\bar{x}\) is the mean of these observations (41).
• \(N\) is the number of observations (5).

Let's calculate the squared difference for each observation:

• For 32: \((32 - 41)^2 = (-9)^2 = 81\)
• For 35: \((35 - 41)^2 = (-6)^2 = 36\)
• For 41: \((41 - 41)^2 = (0)^2 = 0\)
• For 47: \((47 - 41)^2 = (6)^2 = 36\)
• For 50: \((50 - 41)^2 = (9)^2 = 81\)

Now, sum these squared differences:

Sum of squared differences = \(81 + 36 + 0 + 36 + 81 = 234\)

Finally, calculate the variance by dividing the sum of squared differences by the number of observations (\(N=5\)):

\(\sigma^2 = \frac{234}{5}\)

\(\sigma^2 = 46.8\)

Therefore, the variance of the largest five observations is 46.8.