To find the mean, median, and mode from the given frequency data table, we'll use the following formulas:

Step-by-step Solutions:

1. Mean:

The mean (average) is calculated by summing all the data values, each multiplied by their respective frequency, and then dividing by the total number of observations.

$\text{Mean} = \frac{\sum (\text{data} \times \text{freq})}{\sum \text{freq}}$

Let's calculate it:

$\text{Mean} = \frac{(27 \times 1) + (28 \times 4) + (29 \times 2) + (30 \times 3) + (31 \times 10)}{1 + 4 + 2 + 3 + 10}$

$\text{Mean} = \frac{27 + 112 + 58 + 90 + 310}{20}$

$\text{Mean} = \frac{597}{20} = 29.85$

So, the mean is 29.85.

2. Median:

The median is the middle value of the data set when it is ordered. If there is an even number of observations, the median is the average of the two middle numbers. First, let's find the cumulative frequencies:

Data	Frequency	Cumulative Frequency
27	1	1
28	4	5
29	2	7
30	3	10
31	10	20

The total number of observations is 20, which is even. The median will be the average of the 10th and 11th values. Both the 10th and 11th values are 30 (since cumulative frequency at 30 is 10, and at 31 is 20).

Therefore, the median is:

$\text{Median} = 30$

3. Mode:

The mode is the value that appears most frequently in the data set. From the table, the data value with the highest frequency is 31 (with a frequency of 10).

So, the mode is 31.

Summary:

• Mean = 29.85
• Median = 30
• Mode = 31

Would you like further details or have any questions?

Related Questions:

1. What is the range of this data set?
2. How do you calculate the variance and standard deviation of this data set?
3. What is the effect on the mean if one of the frequencies increases?
4. How would you find the quartiles for this data set?
5. How do you interpret the mode in a frequency distribution?

Tip: The median is a good measure of central tendency when the data set has outliers.