$$ \begin{array}{lll} ext { Class freq } & \ 1-10 & 2 & ext { find MAD } \ 11-20 & 2 & ext { sample variance } \ 21-30 & 6 & ext { Standard deviation }\end{array} $$

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$$ \begin{array}{lll}	ext { Class freq } & \ 1-10 & 2 & 	ext { find MAD } \ 11-20 & 2 & 	ext { sample variance } \ 21-30 & 6 & 	ext { Standard deviation }\end{array} $$

UpStudy AI · Accepted Answer

To find the Mean Absolute Deviation (MAD), sample variance, and standard deviation for the given frequency distribution, we first need to calculate the midpoints of each class, the total frequency, the mean, and then use these to find the required statistics.

### Step 1: Calculate Midpoints and Frequencies

1. **Class Intervals and Frequencies:**
   - $$1-10$$: Frequency = 2
   - $$11-20$$: Frequency = 2
   - $$21-30$$: Frequency = 6

2. **Midpoints:**
   - For $$1-10$$: Midpoint = $$\frac{1 + 10}{2} = 5.5$$
   - For $$11-20$$: Midpoint = $$\frac{11 + 20}{2} = 15.5$$
   - For $$21-30$$: Midpoint = $$\frac{21 + 30}{2} = 25.5$$

3. **Total Frequency (N):**
   $$
   N = 2 + 2 + 6 = 10
   $$

### Step 2: Calculate the Mean

The mean ($$\bar{x}$$) can be calculated using the formula:
$$
\bar{x} = \frac{\sum (f \cdot x)}{N}
$$
where $$f$$ is the frequency and $$x$$ is the midpoint.

Calculating $$f \cdot x$$:
- For $$1-10$$: $$2 \cdot 5.5 = 11$$
- For $$11-20$$: $$2 \cdot 15.5 = 31$$
- For $$21-30$$: $$6 \cdot 25.5 = 153$$

Now, sum these values:
$$
\sum (f \cdot x) = 11 + 31 + 153 = 195
$$

Now, calculate the mean:
$$
\bar{x} = \frac{195}{10} = 19.5
$$

### Step 3: Calculate the Mean Absolute Deviation (MAD)

MAD is calculated using the formula:
$$
\text{MAD} = \frac{\sum f |x - \bar{x}|}{N}
$$

Calculating $$|x - \bar{x}|$$:
- For $$1-10$$: $$|5.5 - 19.5| = 14$$
- For $$11-20$$: $$|15.5 - 19.5| = 4$$
- For $$21-30$$: $$|25.5 - 19.5| = 6$$

Now, calculate $$f |x - \bar{x}|$$:
- For $$1-10$$: $$2 \cdot 14 = 28$$
- For $$11-20$$: $$2 \cdot 4 = 8$$
- For $$21-30$$: $$6 \cdot 6 = 36$$

Now, sum these values:
$$
\sum f |x - \bar{x}| = 28 + 8 + 36 = 72
$$

Now, calculate MAD:
$$
\text{MAD} = \frac{72}{10} = 7.2
$$

### Step 4: Calculate Sample Variance and Standard Deviation

Sample variance ($$s^2$$) is calculated using the formula:
$$
s^2 = \frac{\sum f (x - \bar{x})^2}{N - 1}
$$

Calculating $$(x - \bar{x})^2$$:
- For $$1-10$$: $$(5.5 - 19.5)^2 = 196$$
- For $$11-20$$: $$(15.5 - 19.5)^2 = 16$$
- For $$21-30$$: $$(25.5 - 19.5)^2 = 36$$

Now, calculate $$f (x - \bar{x})^2$$:
- For $$1-10$$: $$2 \cdot 196 = 392$$
- For $$11-20$$: $$2 \cdot 16 = 32$$
- For $$21-30$$: $$6 \cdot 36 = 216$$

Now, sum these values:
$$
\sum f (x - \bar{x})^2 = 392 + 32 + 216 = 640
$$

Now, calculate sample variance:
$$
s^2 = \frac{640}{10 - 1} = \frac{640}{9} \approx 71.11
$$

### Step 5: Calculate Standard Deviation

Standard deviation ($$s$$) is the square root of the variance:
$$
s = \sqrt{s^2} = \sqrt{71.11} \approx 8.43
$$

### Summary of Results
- **Mean Absolute Deviation (MAD)**: $$7.2$$
- **Sample Variance**: $$\approx 71.11$$
- **Standard Deviation**: $$\approx 8.43$$
Mean Absolute Deviation (MAD): 7.2, Sample Variance: 71.11, Standard Deviation: 8.43

\( \begin{array}{lll}\text { Class freq } & \\ 1-10 & 2 & \text { find MAD } \\ 11-20 & 2 & \text { sample variance } \\ 21-30 & 6 & \text { Standard deviation }\end{array} \)

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