Consider the following dataset representing the ages of a group of students: [18, 20, 22, 21, 19, 23, 25, 18, 22, 24]. Calculate the variance of the ages
The correct answer and explanation is:
To calculate the variance of the dataset [18,20,22,21,19,23,25,18,22,24][18, 20, 22, 21, 19, 23, 25, 18, 22, 24], we follow these steps:
✅ Step-by-Step Variance Calculation
- Find the Mean (average) of the data:
Mean=18+20+22+21+19+23+25+18+22+2410=21210=21.2\text{Mean} = \frac{18 + 20 + 22 + 21 + 19 + 23 + 25 + 18 + 22 + 24}{10} = \frac{212}{10} = 21.2
- Subtract the Mean from Each Value and Square the Result:
(18−21.2)2=(−3.2)2=10.24(20−21.2)2=(−1.2)2=1.44(22−21.2)2=(0.8)2=0.64(21−21.2)2=(−0.2)2=0.04(19−21.2)2=(−2.2)2=4.84(23−21.2)2=(1.8)2=3.24(25−21.2)2=(3.8)2=14.44(18−21.2)2=(−3.2)2=10.24(22−21.2)2=(0.8)2=0.64(24−21.2)2=(2.8)2=7.84(18 – 21.2)^2 = (-3.2)^2 = 10.24 \\ (20 – 21.2)^2 = (-1.2)^2 = 1.44 \\ (22 – 21.2)^2 = (0.8)^2 = 0.64 \\ (21 – 21.2)^2 = (-0.2)^2 = 0.04 \\ (19 – 21.2)^2 = (-2.2)^2 = 4.84 \\ (23 – 21.2)^2 = (1.8)^2 = 3.24 \\ (25 – 21.2)^2 = (3.8)^2 = 14.44 \\ (18 – 21.2)^2 = (-3.2)^2 = 10.24 \\ (22 – 21.2)^2 = (0.8)^2 = 0.64 \\ (24 – 21.2)^2 = (2.8)^2 = 7.84
- Find the Average of These Squared Differences (This is the variance):
Variance=10.24+1.44+0.64+0.04+4.84+3.24+14.44+10.24+0.64+7.8410=53.610=5.36\text{Variance} = \frac{10.24 + 1.44 + 0.64 + 0.04 + 4.84 + 3.24 + 14.44 + 10.24 + 0.64 + 7.84}{10} = \frac{53.6}{10} = \boxed{5.36}
📘 Explanation (Approx. 300 Words)
Variance is a fundamental statistical concept used to measure the spread or dispersion of a set of numbers. It tells us how far each number in the dataset lies from the mean (average) and, consequently, from every other number in the set.
In this example, we’re given the ages of 10 students. The first step in computing the variance is to calculate the mean—the average age. Once we know the mean, we find how much each individual age deviates from this average. However, since simply summing the deviations would cancel out positive and negative differences, we square each deviation to ensure all values are positive.
After squaring, we sum all these squared differences. Dividing this sum by the total number of observations (in this case, 10) gives us the population variance, which in this example is 5.36. This value tells us that the squared average deviation from the mean age is 5.36 years².
A larger variance would indicate that the data points are more spread out from the mean, while a smaller variance would show they are closer together. In real-world contexts like education or quality control, variance helps assess consistency or predict trends.
Thus, in this case, a variance of 5.36 suggests that while the students’ ages vary somewhat, they are relatively close to the average age of 21.2 years.