Find the arithmetic mean using the assumed mean method:

Class Interval

• 100-120: 10
• 120-140: 20
• 140-160: 30
• 160-180: 15
• 180-200: 5

To find the arithmetic mean using the assumed mean method, we first need to understand the data provided and how to apply this technique effectively. The assumed mean method is particularly useful when dealing with grouped data, as it simplifies calculations by using a midpoint and a deviation from that midpoint.

Step-by-Step Calculation

Let's break down the process into clear steps:

1. Identify the Class Intervals and Frequencies

We have the following class intervals and their corresponding frequencies:

• 100-120: 10
• 120-140: 20
• 140-160: 30
• 160-180: 15
• 180-200: 5

2. Calculate the Midpoints

The midpoint for each class interval is calculated by taking the average of the lower and upper boundaries. Here are the midpoints:

• 100-120: (100 + 120) / 2 = 110
• 120-140: (120 + 140) / 2 = 130
• 140-160: (140 + 160) / 2 = 150
• 160-180: (160 + 180) / 2 = 170
• 180-200: (180 + 200) / 2 = 190

3. Choose an Assumed Mean

For simplicity, we can choose the assumed mean (A) as 150, which is the midpoint of the third class interval (140-160).

4. Calculate Deviations from the Assumed Mean

Next, we calculate the deviation of each midpoint from the assumed mean:

• For 110: 110 - 150 = -40
• For 130: 130 - 150 = -20
• For 150: 150 - 150 = 0
• For 170: 170 - 150 = 20
• For 190: 190 - 150 = 40

5. Multiply Deviations by Frequencies

Now, we multiply each deviation by its corresponding frequency:

• For 110: -40 * 10 = -400
• For 130: -20 * 20 = -400
• For 150: 0 * 30 = 0
• For 170: 20 * 15 = 300
• For 190: 40 * 5 = 200

6. Sum the Frequencies and the Products

Next, we sum the frequencies and the products of deviations:

• Total Frequency (N) = 10 + 20 + 30 + 15 + 5 = 80
• Sum of Products = -400 - 400 + 0 + 300 + 200 = -300

7. Calculate the Mean

Finally, we can calculate the arithmetic mean using the formula:

Mean = A + (Sum of Products / Total Frequency)

Substituting the values we have:

Mean = 150 + (-300 / 80)

Mean = 150 - 3.75 = 146.25

Final Result

The arithmetic mean of the given data using the assumed mean method is 146.25.

This method is particularly useful for grouped data as it simplifies calculations while still providing an accurate measure of central tendency. If you have any further questions or need clarification on any of the steps, feel free to ask!