Fundamental Concepts and Applications in Introductory Statistics

Statistical and Practical Significance

Definitions and Distinctions

Understanding the difference between statistical significance and practical significance is essential in interpreting results from statistical analyses.

• Statistical significance: Achieved when results are very unlikely to occur by chance. It is determined by hypothesis testing and p-values.

• Practical significance: Relates to whether the observed effect is large enough to be meaningful in a real-world context, regardless of statistical significance.

Example

A new drug reduces blood pressure by 1 mmHg with a p-value of 0.001. Statistically significant, but the effect may not be practically significant for patient health.

Bias in Data Sources

Potential for Bias

Bias occurs when a data source or collection method systematically favors certain outcomes.

• Organizations may have incentives to present data in a way that supports their interests, leading to potential bias.

• Independent data sources with no incentive to alter results are less likely to introduce bias.

Example

A car club reporting on public transportation may have a bias if their members prefer driving.

Statistical Analysis of Data

Significance and Practicality

• Statistical significance: Results unlikely due to chance.

• Practical significance: Results are large enough to matter in practice.

Sample Data Analysis

Analyzing temperature data at different times can reveal correlations or differences between groups.

Subject	8 AM	12 AM
1	72	66
2	68	67
3	77	72
4	75	69
5	82	76
6	56	54

Analysis can address whether there is a difference between average temperatures for males and females, or a correlation between times.

Parameters and Statistics

Definitions

• Parameter: A numerical measurement describing a characteristic of a population.

• Statistic: A numerical measurement describing a characteristic of a sample.

Example

The mean income of all residents in a city is a parameter; the mean income of a sample of residents is a statistic.

Types of Data: Qualitative vs. Quantitative

Definitions

• Qualitative data: Describes categories or qualities (e.g., colors, names).

• Quantitative data: Consists of counts or measurements (e.g., height, weight).

Example

The price of gasoline is quantitative; types of gasoline are qualitative.

Discrete vs. Continuous Data

Definitions

• Discrete data: Can take only specific values, often counts (e.g., number of students).

• Continuous data: Can take any value within an interval (e.g., height, weight).

Example

Farm produce lengths are continuous; number of cows is discrete.

Levels of Measurement

Four Levels

• Nominal: Categories only, no order (e.g., social security numbers).

• Ordinal: Categories with order, but no meaningful differences (e.g., rankings).

• Interval: Ordered, differences are meaningful, but no true zero (e.g., temperature in Celsius).

• Ratio: Ordered, meaningful differences, true zero exists (e.g., height, weight).

Level	Example
Nominal	Social security numbers
Ordinal	Class rankings
Interval	Temperature (Celsius)
Ratio	Length in meters

Measures of Center

Mean, Median, Mode, and Midrange

• Mean: The arithmetic average. $\text{Mean} = \frac{\sum x}{n}$

• Median: The middle value when data are ordered.

• Mode: The value that appears most frequently.

• Midrange: The average of the maximum and minimum values. $\text{Midrange} = \frac{\text{Max} + \text{Min}}{2}$

Example

Given data: 72, 68, 77, 75, 82, 56

• Mean: $\frac{72+68+77+75+82+56}{6} = 71.67$

• Median: 73.5

• Mode: None (all values unique)

• Midrange: $\frac{82+56}{2} = 69$

Measures of Variation

Range, Variance, and Standard Deviation

• Range: Difference between maximum and minimum values. $\text{Range} = \text{Max} - \text{Min}$

• Variance: Average squared deviation from the mean. $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

• Standard deviation: Square root of variance. $s = \sqrt{s^2}$

Example

Room prices: 294, 123, 281, 228, 232, 128, 219

• Range: $294 - 123 = 171$

• Standard deviation: $75.3$

• Variance: $5672.0$

Symbols for Statistics and Parameters

Concept	Sample Symbol	Population Symbol
Mean	$\bar{x}$	$\mu$
Standard deviation	$s$	$\sigma$
Variance	$s^2$	$\sigma^2$

Descriptive vs. Inferential Statistics

Definitions

• Descriptive statistics: Summarize or describe characteristics of a data set (e.g., mean, median, mode).

• Inferential statistics: Use sample data to make generalizations about a population.

Measures of Center and Outliers

Choosing the Best Measure

• Median is preferred when data have outliers, as it is less affected by extreme values.

• Mean is sensitive to outliers.

• Mode is useful for categorical data.

Summary of Key Concepts

• Statistical significance does not imply practical significance.

• Bias can arise from data sources with incentives.

• Parameters describe populations; statistics describe samples.

• Data can be qualitative or quantitative, discrete or continuous.

• Levels of measurement: nominal, ordinal, interval, ratio.

• Measures of center: mean, median, mode, midrange.

• Measures of variation: range, variance, standard deviation.

• Descriptive statistics summarize data; inferential statistics generalize to populations.

Additional info: Some explanations and examples have been expanded for clarity and completeness.