Procedure to Find Mean, Mode, and Median for Grouped and Ungrouped Data

1. Mean

The mean is the average of a data set.

a. Ungrouped Data

1. Sum all the values in the data set.
2. Count the number of data points.
3. Divide the sum by the number of data points.

Formula: $\text{Mean} = \frac{\sum x_i}{n}$ Where $x_i$ are the data points and $n$ is the number of data points.

b. Grouped Data

1. Identify the midpoints (class marks) of each class interval.
2. Multiply the midpoints by the corresponding frequencies.
3. Sum these products.
4. Divide the sum by the total frequency.

Formula: $\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}$ Where $f_i$ is the frequency and $x_i$ is the midpoint of the class interval.

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2. Mode

The mode is the value that occurs most frequently in a data set.

a. Ungrouped Data

1. Identify the data value(s) that appear most frequently.
2. The value with the highest frequency is the mode.
   • If multiple values have the same highest frequency, the data is multimodal.

b. Grouped Data

1. Identify the modal class (the class with the highest frequency).
2. Use the following formula to calculate the mode:

Formula: $\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$ Where:

• $L$ is the lower boundary of the modal class.
• $f_1$ is the frequency of the modal class.
• $f_0$ is the frequency of the class preceding the modal class.
• $f_2$ is the frequency of the class succeeding the modal class.
• $h$ is the class width.

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3. Median

The median is the middle value when the data is arranged in ascending order.

a. Ungrouped Data

1. Arrange the data in ascending order.
2. If the number of data points $n$ is odd, the median is the middle value.
3. If $n$ is even, the median is the average of the two middle values.

Formula:

• For odd $n$: $\text{Median} = x_{\left(\frac{n+1}{2}\right)}$
• For even $n$: $\text{Median} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2}$

b. Grouped Data

1. Identify the median class (the class where the cumulative frequency reaches or exceeds half the total frequency).
2. Use the following formula to calculate the median:

Formula: $\text{Median} = L + \left( \frac{\frac{n}{2} - C}{f} \right) \times h$ Where:

• $L$ is the lower boundary of the median class.
• $n$ is the total frequency.
• $C$ is the cumulative frequency of the class preceding the median class.
• $f$ is the frequency of the median class.
• $h$ is the class width.

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Summary

• Mean: Average value of the data.
• Mode: Most frequent value or the class with the highest frequency.
• Median: Middle value of the data when sorted.

Would you like to go into more detail on any part of this procedure? Feel free to ask questions.

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Related Questions

1. How do you calculate the variance and standard deviation for grouped data?
2. What are the advantages of using grouped data over ungrouped data?
3. How do you handle data sets with no mode?
4. Can you explain how to calculate the mean using a frequency distribution table?
5. How does the range of data affect the calculation of mode and median?

Tip: In large data sets, always organize your data first (such as using a frequency distribution) to simplify calculations of mean, mode, and median.