Ch 3 - Central Tendency - student
Chapter 3: Central Tendency
Review of Frequency Distribution Tables
Benefits of Frequency Distribution Tables:
They summarize data, providing a clear view of the distribution of values.
Show the number of occurrences of each value or class interval in a dataset.
Grouped Frequency Distribution Table:
Similar benefits as frequency tables but simplify data into class intervals to make interpretation easier.
Example Measurement:
Measuring daily TV watching time with apparent limits for highest class intervals:
0-59 minutes
60-119 minutes
120-179 minutes
Variable: Minutes of TV watched (Continuous)
Scale of Measurement: Ratio.
Graphing Frequency Distributions
To graph frequency distribution:
Select an appropriate graph (likely a histogram for continuous data).
Interval Width of 60:
Considered good for this dataset as it balances detail and readability, accommodating group statistics effectively.
Introduction to Central Tendency
Definition:
A single score representing the most typical or average value in a dataset.
It defines the center of a distribution.
Type of Statistics: Descriptive statistics.
Problems with Central Tendency
Selection of Values:
No standardized method for calculation leads to variations in central tendency scores.
Different methods yield different results, especially with skewed distributions.
Measuring Central Tendency
Three Methods:
Mean: The average value, calculated as the sum of all scores divided by the number of scores.
Median: The middle value in a sorted list of scores (50% below, 50% above).
Mode: The most frequently occurring score or category in the distribution.
Average Definition:
Often refers to the mean but encompasses mean, median, and mode.
The Mean
Definition:
Arithmetic average, symbolized as:
Ī¼ (population mean) = Ī£X / N
M (sample mean) = Ī£X / n
Characteristics:
Represents the balance point of a distribution.
Example: Equal distribution of items among individuals (e.g., burritos = 6 each).
Calculation Example:
Data: 1, 4, 11, 15, ... Sum all data points and divide by count (n).
Change Factors:
Modifying scores affects Ī£X but not n.
Mean trends toward the direction of changed scores but not proportionally.
Adding or Removing Scores:
Both actions alter Ī£X and n, shifting mean toward or away from the added or removed score.
Weighted Means:
Calculation for populations divided into groups:
Combined mean = (Sum of all scores) / (Total size)
Important in assessing combined scores across different demographics.
Usage and Limitations of the Mean:
Preferred for representing central tendency
Not useful when dealing with extreme scores or nominal/ordinal scales.
The Median
Definition:
Midpoint score in a ranked distribution.
Divides the data into two equal halves.
Calculation Process:
Order data from lowest to highest.
For odd n, find the middle score; for even n, average middle two scores.
When to Use the Median:
Effective when data is skewed or has outliers.
The Mode
Definition:
Most frequently occurring score in a distribution.
Can be calculated through a frequency table.
Change Implications:
Adjusting scores may modify the mode based on frequency shifts.
Impact of Skewed Distributions
Mean vs. Median vs. Mode:
Skews affect the placement of mean, median, and mode differently, revealing insights about the data centrality.
Specific Effects:
Mean: Strongly affected by extreme values.
Median: Mildly impacted.
Mode: Unaffected by skew.
Reporting Central Tendency in Research:
Use standard notation for means (M).
Median and mode lack standardized symbols; common practice includes displaying all together in graphics.
Statistical Integrity Considerations:
Skews can lead to misleading representations of averages in reported data.
Practice Problems:
Engage in deriving the mean, median, and mode from provided datasets for practical understanding.