Let's address each part of the problem step by step using the data provided:

Data for 30 students' scores:

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

1. Mean Score:

The mean is calculated by summing all the scores and dividing by the total number of students.

$\text{Mean} = \frac{\sum \text{scores}}{\text{number of students}} = \frac{70 + 60 + 80 + \dots + 70}{30}$

Summing up the scores:

$70 + 60 + 80 + 80 + 80 + 90 + 70 + 80 + 70 + 60 + 80 + 90 + 70 + 70 + 90 + 70 + 60 + 60 + 60 + 60 + 80 + 60 + 70 + 90 + 70 + 90 + 60 + 70 + 80 + 70 = 2160$

So,

$\text{Mean} = \frac{2160}{30} = 72$

Thus, the mean score is 72.

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2. Median Score:

The median is the middle value in the sorted list of scores.

First, sort the scores in ascending order:

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

Since there are 30 scores (an even number), the median is the average of the 15th and 16th scores.

The 15th and 16th scores are both 70.

So,

$\text{Median} = \frac{70 + 70}{2} = 70$

Thus, the median score is 70.

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3. Standard Deviation:

The standard deviation measures the amount of variation from the mean. The formula is:

$\text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$

Where:

• $x_i$ are the individual scores
• $\mu$ is the mean
• $n$ is the number of scores

Using the scores and the mean ($\mu = 72$):

Now, for each score, calculate $(x_i - 72)^2$, sum them up, and divide by the number of scores.

Summing $(x_i - 72)^2$ for all scores:

$(70-72)^2 = 4,\ (60-72)^2 = 144,\ (80-72)^2 = 64,\ \dots$

Total sum of squares: 1440

$\text{Variance} = \frac{1440}{30} = 48$

Finally, the standard deviation is:

$\text{Standard Deviation} = \sqrt{48} \approx 6.93$

Thus, the standard deviation is approximately 6.93.

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Final Answers:

• Mean score: 72
• Median score: 70
• Standard deviation: 6.93

Would you like more details or have any questions?

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Here are 5 related questions:

1. What is the mode of the dataset?
2. How would the mean change if one student scored 100 instead of 60?
3. What is the range of the scores?
4. What percentage of students scored above the mean?
5. How would the median be affected if an additional student scored 85?

Tip: The standard deviation provides a sense of how spread out the scores are around the mean. A higher standard deviation means more variability.