Question

date
discrete series
example
marks(x) no of student
20 8
30 12
40 20
50 10
60 6
70 4
60

Explanation:

To solve problems related to this discrete series (e.g., calculating mean, median, mode, variance, etc.), let's take the example of calculating the arithmetic mean for the given discrete series of marks and number of students.

Step-by-Step Calculation for Arithmetic Mean

In a discrete series, the arithmetic mean (\(\bar{x}\)) is calculated using the formula:
$$\bar{x} = \frac{\sum (x \cdot f)}{\sum f}$$
where:

• \(x\) = marks (variable),
• \(f\) = number of students (frequency),
• \(\sum (x \cdot f)\) = sum of the product of marks and their respective frequencies,
• \(\sum f\) = total number of students (sum of frequencies).

Step 1: Calculate \(x \cdot f\) for each row

Multiply each mark (\(x\)) by its corresponding number of students (\(f\)):

Marks (\(x\))	No. of Students (\(f\))	\(x \cdot f\)
30	12	\(30 \times 12 = 360\)
40	20	\(40 \times 20 = 800\)
50	10	\(50 \times 10 = 500\)
60	6	\(60 \times 6 = 360\)
70	4	\(70 \times 4 = 280\)

Step 2: Calculate \(\sum (x \cdot f)\)

Sum all the values of \(x \cdot f\):
$$\sum (x \cdot f) = 160 + 360 + 800 + 500 + 360 + 280$$
$$\sum (x \cdot f) = 2460$$

Step 3: Calculate \(\sum f\)

Sum all the frequencies (\(f\)):
$$\sum f = 8 + 12 + 20 + 10 + 6 + 4$$
$$\sum f = 60$$

Step 4: Calculate the arithmetic mean (\(\bar{x}\))

Using the formula:
$$\bar{x} = \frac{\sum (x \cdot f)}{\sum f} = \frac{2460}{60}$$
$$\bar{x} = 41$$

Final Answer (for Arithmetic Mean)

The arithmetic mean of the marks is \(\boldsymbol{41}\).

(Note: If you need to calculate other measures like median, mode, or variance, the steps will differ, but the discrete series format (with \(x\) and \(f\)) remains the foundation.)

Answer:

To solve problems related to this discrete series (e.g., calculating mean, median, mode, variance, etc.), let's take the example of calculating the arithmetic mean for the given discrete series of marks and number of students.

Step-by-Step Calculation for Arithmetic Mean

In a discrete series, the arithmetic mean (\(\bar{x}\)) is calculated using the formula:
$$\bar{x} = \frac{\sum (x \cdot f)}{\sum f}$$
where:

• \(x\) = marks (variable),
• \(f\) = number of students (frequency),
• \(\sum (x \cdot f)\) = sum of the product of marks and their respective frequencies,
• \(\sum f\) = total number of students (sum of frequencies).

Step 1: Calculate \(x \cdot f\) for each row

Multiply each mark (\(x\)) by its corresponding number of students (\(f\)):

Marks (\(x\))	No. of Students (\(f\))	\(x \cdot f\)
30	12	\(30 \times 12 = 360\)
40	20	\(40 \times 20 = 800\)
50	10	\(50 \times 10 = 500\)
60	6	\(60 \times 6 = 360\)
70	4	\(70 \times 4 = 280\)

Step 2: Calculate \(\sum (x \cdot f)\)

Sum all the values of \(x \cdot f\):
$$\sum (x \cdot f) = 160 + 360 + 800 + 500 + 360 + 280$$
$$\sum (x \cdot f) = 2460$$

Step 3: Calculate \(\sum f\)

Sum all the frequencies (\(f\)):
$$\sum f = 8 + 12 + 20 + 10 + 6 + 4$$
$$\sum f = 60$$

Step 4: Calculate the arithmetic mean (\(\bar{x}\))

Using the formula:
$$\bar{x} = \frac{\sum (x \cdot f)}{\sum f} = \frac{2460}{60}$$
$$\bar{x} = 41$$

Final Answer (for Arithmetic Mean)

The arithmetic mean of the marks is \(\boldsymbol{41}\).

(Note: If you need to calculate other measures like median, mode, or variance, the steps will differ, but the discrete series format (with \(x\) and \(f\)) remains the foundation.)