Calculate for me the mean ,median and mode

We need the midpoint ($x_i$) for each class and the cumulative frequency ($CF$). The total frequency $N = \sum f_i = 12+30+9+7+6+4 = 68$.

Class Interval	Frequency ($f_i$)	Midpoint ($x_i$)	$f_i x_i$	Cumulative Frequency ($CF$)
1-5	12	3	36	12
6-10	30	8	240	42
11-15	9	13	117	51
16-20	7	18	126	58
21-25	6	23	138	64
26-30	4	28	112	68
Total	$N=68$		769

The formula for the mean of grouped data is:

$$\bar{x} = \frac{\sum f_i x_i}{N}$$

Substitute the values from the table:

$$\bar{x} = \frac{769}{68}$$ $$\bar{x} \approx 11.3088$$

Rounding to two decimal places, the mean is $\boxed{\text{11.31}}$.

The formula for the median of grouped data is:

$$\text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times w$$

First, find the median class. $N/2 = 68/2 = 34$. The 34th value falls in the class 6-10 (since its cumulative frequency is 42, which is the first $CF$ greater than or equal to 34).

• $L$ (lower boundary of median class) = $6 - 0.5 = 5.5$
• $CF$ (cumulative frequency of class before median class) = 12 (for class 1-5)
• $f$ (frequency of median class) = 30
• $w$ (class width) = $10.5 - 5.5 = 5$

Substitute these values into the formula:

$$\text{Median} = 5.5 + \left( \frac{34 - 12}{30} \right) \times 5$$ $$\text{Median} = 5.5 + \left( \frac{22}{30} \right) \times 5$$ $$\text{Median} = 5.5 + \frac{110}{30}$$ $$\text{Median} = 5.5 + 3.666...$$

Rounding to two decimal places, the median is $\boxed{\text{9.17}}$.

The formula for the mode of grouped data is:

$$\text{Mode} = L + \left( \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \right) \times w$$

First, identify the modal class (the class with the highest frequency). The highest frequency is 30, which corresponds to the class 6-10.

• $L$ (lower boundary of modal class) = $6 - 0.5 = 5.5$
• $f_m$ (frequency of modal class) = 30
• $f_1$ (frequency of class before modal class) = 12 (for class 1-5)
• $f_2$ (frequency of class after modal class) = 9 (for class 11-15)
• $w$ (class width) = $10.5 - 5.5 = 5$

Substitute these values into the formula:

$$\text{Mode} = 5.5 + \left( \frac{30 - 12}{(30 - 12) + (30 - 9)} \right) \times 5$$ $$\text{Mode} = 5.5 + \left( \frac{18}{18 + 21} \right) \times 5$$ $$\text{Mode} = 5.5 + \left( \frac{18}{39} \right) \times 5$$ $$\text{Mode} = 5.5 + \frac{90}{39}$$ $$\text{Mode} = 5.5 + 2.307...$$

Rounding to two decimal places, the mode is $\boxed{\text{7.81}}$.