On 6 quizzes Joe scored 73,88,86,90,87 and 29. What is the mean and standard deviation. On the 7th quiz he scored 32 - how does this affect the mean and standard deviation?

Let's break this down step by step.

### 1. Calculate the Mean and Standard Deviation for the First 6 Quizzes:

The scores for the first 6 quizzes are: 73, 88, 86, 90, 87, and 29.

#### Mean (First 6 Quizzes):

The mean (average) is calculated as the sum of the scores divided by the number of scores:

\[
\text{Mean} = \frac{73 + 88 + 86 + 90 + 87 + 29}{6}
\]

Let's calculate the mean.

#### Standard Deviation (First 6 Quizzes):

The standard deviation measures the spread of the data. First, we calculate the variance using this formula:

\[
\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n}
\]
Where:
- \( x_i \) are the individual scores
- \( \bar{x} \) is the mean
- \( n \) is the number of scores

The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]

### 2. Effect of the 7th Quiz Score:

Now, Joe scores 32 on the 7th quiz. We need to calculate the new mean and the new standard deviation including this score.

- The new mean will be calculated with 7 scores.
- The new standard deviation will also be recalculated considering all 7 scores.

I'll perform both sets of calculations.### Results:

1. **Mean and Standard Deviation for the First 6 Quizzes:**
   - Mean: \( 75.5 \)
   - Standard Deviation: \( 21.52 \)

2. **After the 7th Quiz Score (32):**
   - New Mean: \( 69.29 \)
   - New Standard Deviation: \( 25.07 \)

### Effect of the 7th Quiz:
- The **mean** decreased from 75.5 to 69.29, showing that the lower score of 32 pulled the average down.
- The **standard deviation** increased from 21.52 to 25.07, indicating that the spread of the scores has become larger due to the addition of the new outlier (32).

Would you like further details or have any questions?

Here are 5 questions to consider next:
1. How would removing the highest score affect the mean and standard deviation?
2. What happens to the mean and standard deviation if the score 29 was replaced with 50?
3. How is the standard deviation affected by scores that are very close to the mean?
4. How would adding another low score, say 20, change the mean and standard deviation?
5. What is the difference between population and sample standard deviation?

**Tip:** A larger standard deviation indicates greater variability in your dataset.

On 6 quizzes Joe scored 73,88,86,90,87 and 29. What is the mean and standard deviation? On the 7th quiz he scored 32 - how does this affect the mean and standard deviation?

Analyze how the mean and standard deviation of Joe's quiz scores (73, 88, 86, 90, 87, 29) change when a new score of 32 is added. Learn the impact of outliers on statistical measures in this detailed example. Suitable for Grades 9-11.

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