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Animated Solution for Mathematics - Statistics: In an experiment with 15 observations on x, the following results were available: $\sum x^{2} = 2830, \sum x = 170$ . One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

Select Answer:

Progressive Hints

Hint 1

Hint 1: The Spark. Statistics is about the collective behavior of data. When you change a single observation, you are not just changing one number; you are altering the global properties of the entire set. Think of the sum of observations and the sum of squares as the DNA of your dataset. You must update these DNA markers before you can calculate the new variance.

Hint 2

Hint 2: The Arsenal. Remember the fundamental relationship between variance, the sum of squares, and the mean:

σ^{2} = \frac{\sum x ^{2}}{n} - (\overset{x}{ˉ})^{2}

. The trap here is that

\sum x^{2}

is not the same as

(\sum x)^{2}

. You need to update the sum of observations

\sum x

and the sum of squares

\sum x^{2}

separately using the replacement logic.

Hint 3

Hint 3: The Execution. To find the corrected values, use these specific update rules:

\sum x_{n e w} = \sum x_{o l d} - x_{w r o n g} + x_{cor r ec t}

and

\sum x_{n e w}^{2} = \sum x_{o l d}^{2} - (x_{w r o n g})^{2} + (x_{cor r ec t})^{2}

. Once you have these new sums, calculate the new mean and plug everything into the variance formula.

Visualized Solution

Identify Given Parameters

Number of observations $n = 15$
Given incorrect sum $\sum x = 170$
Given incorrect sum of squares $\sum x^{2} = 2830$

The Error Detection

Incorrect value $= 20$
Correct value $= 30$

Variance Formula Bridge

Variance formula: $σ^{2} = \frac{\sum x ^{2}}{n} - (\frac{\sum x}{n})^{2}$
We need corrected values for $\sum x$ and $\sum x^{2}$ .

Correcting the Sum $\sum x$

Corrected $\sum x = (Old \sum x) - (Wrong Value) + (Correct Value)$
Corrected $\sum x = 170 - 20 + 30$

Calculating Corrected $\sum x$

Corrected $\sum x = 150 + 30$
Corrected $\sum x = 180$

Correcting Sum of Squares $\sum x^{2}$

Corrected $\sum x^{2} = (Old \sum x^{2}) - (Wrong Value)^{2} + (Correct Value)^{2}$
Corrected $\sum x^{2} = 2830 - 2 0^{2} + 3 0^{2}$

Calculating Corrected $\sum x^{2}$

Corrected $\sum x^{2} = 2830 - 400 + 900$
Corrected $\sum x^{2} = 3330$

Calculating Corrected Mean

Corrected Mean $\overset{x}{ˉ} = \frac{Corrected \sum x}{n}$
$\overset{x}{ˉ} = \frac{180}{15} = 12$

Final Variance Setup

Corrected Variance $= \frac{Corrected \sum x ^{2}}{n} - (Corrected Mean)^{2}$
Corrected Variance $= \frac{3330}{15} - (12)^{2}$

Execution of Division

$\frac{3330}{15} = 222$
$1 2^{2} = 144$

The Final Result

Corrected Variance $= 222 - 144$
Corrected Variance $= 78$

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The Sigma Insight: Measures of Dispersion

Solution Diagram

Analyzing the Setup

In this statistical problem, we are given

n = 15

observations. The initial sum of these observations is

\sum x = 170

, and the initial sum of their squares is

\sum x^{2} = 2830

A data point was incorrectly recorded as

20

instead of

30

. To find the correct variance, we must first adjust these two fundamental pillars of the dataset.

The Logic of Correction

We begin by correcting the sum of the observations. We subtract the erroneous value and add the correct value to the original sum:

\sum x_{n e w} = 170 - 20 + 30 = 180

Next, we correct the sum of the squares. We must subtract the square of the incorrect value and add the square of the correct value:

\sum x_{n e w}^{2} = 2830 - (20)^{2} + (30)^{2}

Performing the arithmetic, we obtain:

\sum x_{n e w}^{2} = 2830 - 400 + 900 = 3330

The Final Calculation

With the corrected sums, we first calculate the corrected mean, denoted as

\overset{x}{ˉ}

\overset{x}{ˉ} = \frac{180}{15} = 12

Now, we apply the standard formula for variance,

σ^{2} = \frac{\sum x ^{2}}{n} - (\overset{x}{ˉ})^{2}

. Substituting our updated values into the equation:

σ^{2} = \frac{3330}{15} - (12)^{2}

Calculating the division and the square, we find:

σ^{2} = 222 - 144 = 78

The corrected variance of the dataset is $78$ .

Common Traps & FAQs

Wait, why can't I just calculate the variance of the old data and then add the difference between the wrong and correct values?

That is a classic trap! Variance is a non-linear measure. It depends on the square of the values. Because of this, you cannot simply add or subtract a constant difference to the final variance. You must go back to the raw sums, update them, and then recompute the variance from scratch.

Do I need to change the number of observations

n

when I replace a value?

No, you do not. Since you are replacing one value with another, the total count of observations remains exactly the same.

n

stays

15

throughout the process.

Why is the variance formula

\frac{\sum x ^{2}}{n} - (\frac{\sum x}{n})^{2}

instead of

\frac{\sum ( x - x ˉ ) ^{2}}{n}

Both formulas are mathematically equivalent! The first one, often called the computational formula, is much easier to use when you are given

\sum x

and

\sum x^{2}

directly. It saves you from having to calculate the deviation of every single point from the mean.

Animated Solution for Mathematics - Statistics: In an experiment with 15 observations on x, the following results were available: $\sum x^{2} = 2830, \sum x = 170$ . One observation that was 20 was found to be wrong and was replaced by the correct value 30. The corrected variance is

Select Answer:

Visualized Solution

Identify Given Parameters

The Error Detection

Variance Formula Bridge

Correcting the Sum $\sum x$

Calculating Corrected $\sum x$

Correcting Sum of Squares $\sum x^{2}$

Calculating Corrected $\sum x^{2}$

Calculating Corrected Mean

Final Variance Setup

Execution of Division

The Final Result

The Sigma Insight: Measures of Dispersion

Analyzing the Setup

The Logic of Correction

The Final Calculation

Similar Questions

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Select Answer:

Visualized Solution

Identify Given Parameters

The Error Detection

Variance Formula Bridge

Correcting the Sum .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑x

Calculating Corrected .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑x

Correcting Sum of Squares .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑x2

Calculating Corrected .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑x2

Calculating Corrected Mean

Final Variance Setup

Execution of Division

The Final Result

The Sigma Insight: Measures of Dispersion

Analyzing the Setup

The Logic of Correction

The Final Calculation

Similar Questions

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Correcting the Sum $\sum x$

Calculating Corrected $\sum x$

Correcting Sum of Squares $\sum x^{2}$

Calculating Corrected $\sum x^{2}$

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