Let's solve the three questions step by step using the data from the table.

1. Mean Score for the Exam

The mean score is calculated as:

$\text{Mean} = \frac{\text{Sum of all scores}}{\text{Number of scores}}$

Here, the table contains 30 scores. Let's calculate the total sum and the mean.

• Scores in the table are: 90, 90, 90, 60, 80, 60, 60, 80, 60, 80, 80, 90, 90, 60, 90, 90, 80, 60, 60, 60, 60, 80, 90, 80, 80, 90, 60, 60, 70, 80.

Sum of all scores:

$90 + 90 + 90 + 60 + 80 + 60 + 60 + 80 + 60 + 80 + 80 + 90 + 90 + 60 + 90 + 90 + 80 + 60 + 60 + 60 + 60 + 80 + 90 + 80 + 80 + 90 + 60 + 60 + 70 + 80 = 2400$

$\text{Mean} = \frac{2400}{30} = 80$

So, the mean score is 80.

2. Median Score for the Exam

The median is the middle value when the data set is arranged in ascending order.

Arranging the scores in increasing order:

$60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 70, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 90, 90, 90, 90, 90, 90, 90, 90, 90$

There are 30 data points, so the median is the average of the 15th and 16th values. Both values are 80, so:

$\text{Median} = \frac{80 + 80}{2} = 80$

Thus, the median score is 80.

3. Standard Deviation for the Exam

The formula for standard deviation is:

$\text{Standard Deviation} = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \text{mean})^2}$

Where $x_i$ are the individual scores, the mean is 80, and $N = 30$.

Let me calculate the standard deviation.

First, calculate the squared differences from the mean:

$(90 - 80)^2 = 100$ $(80 - 80)^2 = 0$ $(60 - 80)^2 = 400$ $(70 - 80)^2 = 100$

I'll now compute the full sum of squared differences and divide by $N$, then take the square root to find the standard deviation:

Sum of squared differences = 4000

Now, standard deviation:

$\text{Standard Deviation} = \sqrt{\frac{4000}{30}} \approx \sqrt{133.33} \approx 11.55$

Thus, the standard deviation is approximately 11.55.

If you'd like more details or clarifications, feel free to ask! Here are five related questions for further understanding:

1. How would the mean change if one additional student scored 100?
2. What happens to the standard deviation if all scores are increased by 5?
3. How would you calculate the mode from this data set?
4. Why is the median sometimes preferred over the mean?
5. How does the standard deviation help in understanding the spread of the scores?

Tip: The standard deviation gives you an idea of how much the scores deviate from the mean. A small standard deviation means most scores are close to the mean, while a large standard deviation indicates more variability.