Notes on Measures of Tendency and Dispersion from Grouped Data and Weighted Mean

3.3 Measures of Tendency and Dispersion from Grouped Data

When data is grouped into classes with known frequencies or presented as a histogram, we can estimate the mean ($\mu$) and standard deviation ($\sigma$).

Finding the Mean from Grouped Data

1. Find the midpoint ($x<em>i$) of each class:
   $x</em>i = \frac{\text{lower class limit + upper class limit}}{2}$

2. Multiply each midpoint value ($x<em>i$) by its corresponding frequency ($f</em>i$) and sum them up:
   $\sum<em>{i=1}^{N} x</em>i f<em>i = x</em>1 f<em>1 + x</em>2 f<em>2 + … + x</em>N f_N$

3. Find the sum of all the frequencies:
   $\sum<em>{i=1}^{N} f</em>i = f<em>1 + f</em>2 + … + f_N$

4. Calculate the mean ($\mu$):
   $\mu = \frac{\text{sum of the products of midpoints and corresponding frequencies}}{\text{sum of all frequencies}} = \frac{\sum<em>{i=1}^{N} x</em>i f<em>i}{\sum</em>{i=1}^{N} f_i}$

   • Note: For a sample with $n$ data points where n < N, the sample mean $\bar{x}$ is calculated using the same formula but with $N$ replaced by $n$.

Standard Deviation for Grouped Data

The standard deviation is calculated similarly to section 3.2, but here, $x<em>i$ represents the class midpoint and $f</em>i$ is the corresponding frequency.

• Population Standard Deviation
  $\sigma = \sqrt{\frac{\sum<em>{i=1}^{N}(x</em>i - \mu)^2 f<em>i}{\sum</em>{i=1}^{N} f<em>i}} = \sqrt{\frac{(x</em>1 - \mu)^2 f<em>1 + … + (x</em>N - \mu)^2 f<em>N}{f</em>1 + … + f_N}}$

• Sample Standard Deviation
  $s= \sqrt{\frac{\sum<em>{i=1}^{N}(x</em>i - \bar{x})^2 f<em>i}{(\sum</em>{i=1}^{N} f<em>i) -1}} = \sqrt{\frac{(x</em>1 - \bar{x})^2 f<em>1 + … + (x</em>N - \bar{x})^2 f<em>N}{(f</em>1 + … + f_N) -1}}$

Weighted Mean

The weighted mean ($x_w$) of a variable is calculated using:

$x<em>w = \frac{w</em>1x<em>1 + w</em>2x<em>2 + … + w</em>Nx<em>N}{w</em>1 + w<em>2 + … + w</em>N}$

Where $x<em>i$ is each data point and $w</em>i$ is its corresponding weight.

Example: GPA Calculation

Suppose a student earns an A in a 3-credit hour course, a B in a 4-credit hour course, and a D in a 5-credit hour course. Calculate the GPA.

• A = 4 points
• B = 3 points
• C = 2 points
• D = 1 point
• F = 0 points

Practice Problems and Solutions

1. Problem: Approximate the mean weekly grocery bill for the following data:

   Bill (in dollars)	Frequency
   135-139	10
   140-144	13
   145-149	16
   150-154	18
   155-159	11

   Solution: (The solution is multiple choice, the exact steps to solve were not provided.)

2. Problem: Determine the mean temperature of a sample of 100 patients with the following temperature distribution:

   Temperatures	Frequency
   95.6-96.49	1
   96.5-97.39	3
   97.4-98.29	19
   98.3-99.19	28
   99.2-100.09	35
   100.1-100.99	12
   101.0-101.89	2

   Solution: (The solution is multiple choice, the exact steps to solve were not provided.)

3. Problem: A student has test scores of 62, 83, and 91. The term project score is 88, and the homework score is 76. Each test is worth 20% of the final grade, the term project is 25%, and homework is 15%. What is the student's mean score in the class?

   Solution: The problem requires calculating a weighted mean.

4. Problem: Jenny and Kevin make a trail mix with 3 pounds of dried fruit at $4.00 per pound, 2 pounds of nuts at $3.50 per pound, and 5 pounds of granola at $5.00 per pound. Determine the cost per pound of the mix.

   Solution: The problem requires calculating a weighted average.

5. Problem: Approximate the sample standard deviation of monthly telephone bills:

   Bill (in $)	Frequency
   50-52	2
   53-55	5
   56-58	12
   59-61	19
   62-64	7

   Solution: (The solution is multiple choice, the exact steps to solve were not provided.)

Homework: p. 152 # 3-7, 15, 16