MATH 225 Week 3 Discussion – Central Tendency and Variation

Grace J.
August 7, 2025August 7, 2025
CU BSN, MATH-225

Student Name

Chamberlain University

MATH-225 Statistical Reasoning for the Health Sciences

Prof. Name

Date

Measures of Central Tendency

Measures of central tendency are numerical values used to describe the center or average of a data set. They help in summarizing a large amount of data with a single representative value. The three primary measures are the mean, median, and mode. Each provides different insights into the distribution of the data.

What is the Mean?

The mean, often referred to as the average, is the sum of all the data entries divided by the number of entries. It provides a central value that reflects the overall data trend.

Example:
To find the mean of the data set: 10, 7, 15, 6, 24, 20, 1
Sum = 83, Number of entries = 7
Mean = 83 ÷ 7 = 11.857

Another example involves the heights of players on the 2009–2010 Cleveland Cavaliers basketball team, where the mean height is 79.53 inches.

What is the Weighted Mean?

The weighted mean accounts for data entries that contribute differently to the final average, based on assigned weights.

Formula:

xˉ=∑(x⋅w)∑w\bar{x} = \frac{\sum (x \cdot w)}{\sum w}

Where x is the data value and w is the weight.

Example:
In a course where grades are weighted as follows:

50% tests,
15% midterm,
20% final,
10% lab,
5% homework.

With scores:

Tests = 86
Midterm = 96
Final = 98
Lab = 98
Homework = 100

Calculation Table:

Category	Weight	Grade	Weight × Grade
Test	0.50	86	43.0
Midterm	0.15	96	14.4
Final Exam	0.20	98	19.6
Lab Work	0.10	98	9.8
Homework	0.05	100	5.0
Total	1.00	—	91.8

Weighted Mean = 91.8 ÷ 1 = 91.8

What is the Mode?

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

Example:
Data set: 12, 11, 20, 11, 11, 6, 19, 11
Mode = 11 (It appears 4 times, more than any other value.)

What is the Median?

The median is the middle value of an ordered data set.

If the number of data entries is odd, the median is the center value.
If the number of data entries is even, it is the average of the two middle values.

Examples:

Data: 15, 11, 10, 21, 13 → Ordered: 10, 11, 13, 15, 21
Median = 13
Data: 872, 397, 427, 388, 782, 397 → Ordered: 388, 397, 397, 427, 782, 872
Middle values: 397 and 427
Median = (397 + 427) ÷ 2 = 412

Comparing the Mean, Median, and Mode

Each measure represents the center of a data set in a different way.

Measure	Description	Advantage	Disadvantage
Mean	Average value of all entries	Takes into account every value	Affected by outliers
Median	Middle value in ordered data	Resistant to outliers	Ignores exact values
Mode	Most frequent value	Easy to identify	May not be unique

Example: Salaries of 40 dancers

Salary ($)	Number of Dancers
19,000	2
22,000	4
27,000	1
35,000	3
38,000	13
44,000	5
48,000	7
75,000	4
150,000	1

From the data:

Mean salary = $43,950
Median salary = $38,000
Mode = $38,000 (occurs 13 times)

Skewness and Standard Deviation

What is the Range?

The range is the difference between the highest and lowest values in the data set.
Formula:

Range=Maximum−Minimum\text{Range} = \text{Maximum} – \text{Minimum}

How to Calculate Variance?

Population Variance Steps:

Compute the mean (μ)
Subtract the mean from each data value (x – μ)
Square each result
Sum the squared deviations
Divide the sum by the number of data points (N)

Example Data Set: 22, 10, 6, 9, 8

Mean = (22 + 10 + 6 + 9 + 8) / 5 = 11
Range = 22 – 6 = 16