What is Variance for Ungrouped Data?

S7-SA3-0083

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Variance for ungrouped data tells us how spread out a set of numbers is from its average (mean). It measures the average of the squared differences from the mean, giving us a clear idea of data dispersion.

Simple Example

Quick Example

Imagine your cricket team's scores in 5 matches: 10, 90, 50, 40, 60. The average score might be 50. Variance helps us see if scores were consistently around 50 (low variance) or if they varied a lot, like one very low and one very high score (high variance).

Worked Example

Step-by-Step

Let's find the Variance for the daily chai prices (in Rupees) at a small stall over 5 days: 10, 12, 8, 10, 15.

1. **Find the Mean (Average):** Add all the values and divide by the count.
Mean (x̄) = (10 + 12 + 8 + 10 + 15) / 5 = 55 / 5 = 11

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2. **Subtract the Mean from each data point:** (x - x̄)
(10 - 11) = -1
(12 - 11) = 1
(8 - 11) = -3
(10 - 11) = -1
(15 - 11) = 4

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3. **Square each difference:** (x - x̄)^2
(-1)^2 = 1
(1)^2 = 1
(-3)^2 = 9
(-1)^2 = 1
(4)^2 = 16

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4. **Sum these squared differences:** Σ(x - x̄)^2
1 + 1 + 9 + 1 + 16 = 28

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5. **Divide the sum by the total number of data points (N):** For population variance, divide by N. For sample variance, divide by (N-1).
Here, we'll use N (assuming it's the entire data set we care about).
Variance (σ^2) = Σ(x - x̄)^2 / N = 28 / 5 = 5.6

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**Answer:** The Variance is 5.6.

Why It Matters

Variance is super important in fields like AI/ML to understand how reliable predictions are, or in FinTech to measure investment risk. Engineers use it to check consistency in manufacturing, and doctors in Biotechnology use it to study drug effectiveness. Learning this helps you build a strong base for future careers!

Common Mistakes

MISTAKE: Forgetting to square the differences before summing them. | CORRECTION: Always square (x - x̄) values; this makes all values positive and emphasizes larger deviations.

MISTAKE: Confusing population variance (divide by N) with sample variance (divide by N-1). | CORRECTION: For most introductory problems, especially when the data set is small and represents the 'whole picture', divide by N. If it's a 'sample' taken from a larger group, use N-1.

MISTAKE: Calculating the mean incorrectly. | CORRECTION: Double-check your addition and division when finding the mean, as any error here will make all subsequent steps wrong.

Practice Questions

Try It Yourself

QUESTION: Find the Variance for the number of samosas sold per hour: 5, 7, 6, 8, 9. | ANSWER: Mean = 7. Sum of squared differences = 10. Variance = 10 / 5 = 2.

QUESTION: A student's scores in 4 tests are 60, 70, 80, 90. Calculate the Variance. | ANSWER: Mean = 75. Sum of squared differences = 500. Variance = 500 / 4 = 125.

QUESTION: The daily auto-rickshaw fares (in Rupees) for a week were 40, 50, 30, 60, 40, 50, 70. What is the Variance of these fares? | ANSWER: Mean = 48.57 (approx). Sum of squared differences = 1428.57 (approx). Variance = 1428.57 / 7 = 204.08 (approx).

MCQ

Quick Quiz

What does a high variance value indicate about a set of data?

The data points are very close to the mean.

The data points are spread out far from the mean.

The mean of the data is very high.

The data has no outliers.

The Correct Answer Is:

A high variance means the numbers in the data set are far from their average, indicating a wide spread. Low variance means they are clustered closely around the average.

Real World Connection

In the Real World

In cricket, commentators often talk about a batsman's 'consistency'. If a player scores 100, 5, 90, 10, 80, their scores have high variance. A player scoring 50, 55, 45, 60, 50 has low variance and is more consistent. Data analysts use variance to evaluate player performance or predict match outcomes for platforms like Dream11.

Key Vocabulary

Key Terms

MEAN: The average of all numbers in a data set. | DEVIATION: The difference between a data point and the mean. | SQUARED DIFFERENCE: The result of squaring the deviation. | DISPERSION: How spread out the data points are. | UNGROUPED DATA: Raw data not organized into categories or intervals.

What's Next

What to Learn Next

Great job understanding Variance! Next, you should learn about 'Standard Deviation'. It's closely related to variance and is even more commonly used because it's in the same units as your original data, making it easier to interpret.