Math Problem Statement

Class Interva2.docx

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solve for the mean, ungrouped and grouped data

Solution

The mean (or average) is a measure of central tendency that represents the sum of all values divided by the number of values. The method of calculating the mean differs for ungrouped and grouped data.


1. Mean for Ungrouped Data

For ungrouped data (raw data), the mean is calculated using the formula:

xˉ=xin\bar{x} = \frac{\sum x_i}{n}

Where:

  • xˉ\bar{x} = mean
  • xix_i = each individual data value
  • nn = number of values in the dataset

Example 1:

Find the mean of the numbers: 5, 8, 12, 10, 15.

xˉ=5+8+12+10+155=505=10\bar{x} = \frac{5 + 8 + 12 + 10 + 15}{5} = \frac{50}{5} = 10

The mean is 10.


2. Mean for Grouped Data

For grouped data, the mean is calculated using:

xˉ=fxf\bar{x} = \frac{\sum f x}{\sum f}

Where:

  • ff = frequency of each class
  • xx = midpoint of each class (found by lower limit+upper limit2\frac{\text{lower limit} + \text{upper limit}}{2})

Example 2:

Find the mean for the following frequency distribution:

Class IntervalFrequency (ff)
1 - 53
6 - 105
11 - 154
  1. Find the midpoints (xx):

    Midpoint=Lower Limit+Upper Limit2\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}

    Class IntervalFrequency (ff)Midpoint (xx)fxfx
    1 - 531+52=3\frac{1+5}{2} = 33 × 3 = 9
    6 - 1056+102=8\frac{6+10}{2} = 85 × 8 = 40
    11 - 15411+152=13\frac{11+15}{2} = 134 × 13 = 52
  2. Compute the mean:

    xˉ=fxf=9+40+523+5+4=10112=8.42\bar{x} = \frac{\sum fx}{\sum f} = \frac{9 + 40 + 52}{3 + 5 + 4} = \frac{101}{12} = 8.42

The mean is 8.42.


Do you need further details or have any questions?

Here are some related questions to deepen your understanding:

  1. How is the mean different from the median and mode?
  2. How do you handle missing values when calculating the mean?
  3. What are the advantages and disadvantages of using the mean?
  4. How do you calculate the mean from a histogram?
  5. What happens to the mean if an outlier is added to the data?

Tip: In grouped data, always ensure midpoints are correctly calculated before computing the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Measures of Central Tendency
Mean Calculation

Formulas

Mean for Ungrouped Data: x̄ = (Σxi) / n
Mean for Grouped Data: x̄ = (Σfx) / (Σf)

Theorems

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Suitable Grade Level

Grades 6-10