Chapter 10: Basic Statistical Concepts: Descriptive vs Inferential Statistics
Descriptive vs Inferential Statistics
Descriptive statistics: Used to characterize a group based on data taken from the group.
Central tendency: the extent to which data clusters around a point. (The best statistical means to describe an entire group of data with one number)
Variability: the extent to which data are spread out.
Aim: summarize the data with a small set of numbers.
Inferential statistics: Designed to generalize findings from a sample to a larger population; address relationships between variables and differences across conditions.
Levels of Measurement and Data Types
Qualitative (Parametric) vs Quantitative (Nonparametric)
Nominal: Categorical data; Mutually exclusive categories (e.g., gender, marital status, favorite color, college majors).
Ordinal: Categorical data; Ordered categories where distances between categories are not necessarily equal (e.g., levels of satisfaction, letter grades, military rank).
Interval: Differences between data points are meaningful with equal intervals, but no true zero (e.g., SAT scores, IQ scores).
Ratio: Interval data with a true zero, allowing meaningful ratios (e.g., age, height, weight, distance).
Parametric vs Nonparametric data
Parametric: Assumes data come from a population described by parameters (e.g., normal distribution).
Nonparametric: Does not assume a specific distribution (often used for ordinal/nominal data).
Summary mapping (typical for choosing summary statistics):
Nominal: Mode best describes most typical value.
Ordinal: Median best describes central tendency; mode can also be informative.
Interval-Ratio: Mean best describes central tendency under normality assumptions.
Inference groundwork: Many inferential methods assume a normal distribution of the variable of interest.
Measures of Central Tendency (Descriptive)
Mean: arithmetic average of a group of numbers calculated as the sum of all numbers divided by the total number of values in the
Mean= sum of all scores/total number of scores
The mean of a population is represented by the Greek letter μ while the mean of a sample is represented by X.
Characteristics:
Most commonly used measure of central tendency.
Requires numerical values on an interval or ratio scale.
Sensitive to extreme values (outliers).
Example (from transcript): Data: 1, 5, 4, 3, 2
Sum of the score 1+5+4+3+2 =15
Divide the sum (15) by the number of scores (n = 5)
Mean = 15/5=3
x=3
Notes:
Changing any score changes the mean.
In skewed distributions, mean can be pulled toward the tail.
Median: middle value when data are ordered.
For odd n: middle value; for even n: average of the two middle values.
Advantages:
Less affected by extreme scores than the mean.
Suitable for ordinal data and robust in skewed distributions.
In ordinal/nominal data, the median may be less informative or not applicable depending on scale.
Mode: most frequently occurring value.
Useful for nominal data (only measure that makes sense for nominal).
Limitations:
Not informative in bimodal distributions or very small samples.
Not ideal for interval/ratio data when other measures are available.
Note from transcript: Mode is not a very useful measure of central tendency in some contexts.
Summary relationship:
Nominal: Mode
Ordinal: Median (often preferred), can use Mode
Interval/Ratio: Mean (often preferred, especially with normally distributed data)
Typical relationship in symmetric distributions: Mode = Median = Mean (when perfectly normal and symmetric).
Shape of Distributions and When to Use Each Measure
Skewness and Kurtosis:
Kurtosis: vertical shift in the normal curve; middle of the curve can be elevated or depressed.
Skewness: asymmetry in the distribution; highest frequencies not centered.
Positive skewness: tail extends to the right; mean tends to be greater than the median.
Negative skewness: tail extends to the left; mean tends to be less than the median.
Why it matters: Inferential statistics rely on normality assumptions; deviations affect the choice of summary statistics and tests.
Why distributions matter: Inferential statistics assume a normal distribution for many tests; understanding skewness/kurtosis informs data transformation or nonparametric alternatives.
The Normal Distribution and Z-scores
Normal distribution (bell curve): many statistical methods assume data are normally distributed.
Standard Normal Distribution: Z-scores convert raw scores to a common scale.
Z = (X - μ) / σ, where μ is the population mean and σ is the population standard deviation.
Percentile rules (empirical rule):
Within ±1 SD: ~68%
Within ±2 SD: ~95%
Within ±3 SD: ~99%
Z-scores relate to percentages on the standard normal curve (as illustrated in the transcripts).
Z-tables and percentile interpretations are used to determine probabilities and critical values in hypothesis testing.
Standard Normal Distribution details (as shown in slides):
Probability density and cumulative percentages corresponding to Z-scores from -3 to +3.
Relationship between Z-scores and percentiles: e.g., Z = 0 corresponds to 50% percentile, etc.
Measures of Variability (Spread)
Range: difference between the highest and lowest scores.
Formula: ext{Range} = x{ ext{max}} - x{ ext{min}}
Pros: simple; gives a sense of spread.
Cons: highly sensitive to outliers; provides no information about distribution around the center.
Example from transcript: Data set 7,5,9,2,5,9,4,5,3,7,6,10,3,8,4,6,5,4 -> Range = 10 - 2 = 8.
Interquartile Range (IQR): spread of the middle 50% of the data (between Q1 and Q3).
Formula: ext{IQR} = Q3 - Q1
Useful for interval-ratio data and robust to outliers.
Example in transcript: Given ordered data, Q1 = 4, Q3 = 7, so IQR = 3.
Variance and Standard Deviation:
Variance (sample): s^2 = rac{ extstyle sum (x_i - ar{X})^2}{n-1}
Standard Deviation: s =
sqrt{s^2}Interpretation:
Variance measures the average squared deviation from the mean.
Standard deviation measures the average distance of scores from the mean.
Higher variance means scores are more spread out; lower variance means clustering around the mean.
Standard Error of the Mean (SEM): the standard deviation of the sampling distribution of the sample mean.
Formula: SE = rac{s}{ sqrt{n}}
Purpose: assesses how well the sample mean estimates the population mean.
Distinguish SD (spread of individual scores) from SEM (spread of sample means).
Worked example (from transcript, simplified):
Data: 12, 11.7, 10, etc. (illustrates calculation of mean, SD, and SEM).
Final values in one example: Mean = 11.7, SD = 2.66, SEM = 0.841 when n = 10.
Population distribution vs sample statistics:
SEM decreases as n increases, reflecting more precision with larger samples.
The standard deviation (SD) reflects variability in the population/sample and does not inherently change with sample size in the same way SEM does.
Confidence Intervals and Statistical Inference Basics
Confidence Interval (CI): a range computed from the sample that, with a certain level of confidence (e.g., 95%), is expected to contain the true population parameter.
95% CI interpretation: In 95% of samples, the interval would contain the true population mean (or proportion).
Common formula (for a mean with known or large-sample SD):
ext{CI} = ar{X} B1 z{ rac{lpha}{2}} imes SE where SE = rac{s}{ sqrt{n}} and z{ rac{lpha}{2}} is the critical value from the standard normal distribution (e.g., 1.96 for 95%).
Confidence intervals are often reported in research articles and are based on the Standard Error of the Mean.
Confidence intervals example in the transcript: A 95% CI for a mean reported as using the SE and z = 1.96; Lower and upper limits are computed accordingly.
Standard Error of the Mean (SEM) vs Population distribution:
The SEM is the SD of the sampling distribution of the mean.
It estimates how much sample means would vary if we repeated the study many times.
The Normal distribution and sampling: the CI construction relies on normal approximation or t-distribution depending on sample size and whether population SD is known.
Inferential Statistics: Overview and Questions
Purpose: Inferential statistics are designed to generalize from a sample to a larger population.
Key questions:
1) Is there a relationship between variables? (relationships between variables)
2) Is there a difference across conditions? (differences between groups or treatments)
Examples from transcript:
Is there a relationship between GRE scores and success in a graduate program?
Is there a relationship between a field test and a lab-based measure for a variable of interest?
Is there a difference between age groups, gender, training status, etc.?
Is there a difference before and after an intervention?
Hypotheses in inferential statistics:
Null hypothesis (HO): there is no relationship or no difference.
Research/alternative hypothesis (HA): there is a relationship or a difference; may be one-sided or two-sided.
In practice, the hypothesis that is tested is HO; researchers seek to disprove HO.
Errors in hypothesis testing:
Type I error (α): falsely concluding there is a relationship or difference when there is none (false positive).
Type II error (β): falsely concluding there is no relationship or difference when one exists (false negative).
Power = 1 − β: probability of correctly rejecting HO when it is false; researchers often aim for power around 0.80.
Truth table concept: decisions about HO true/false and accept/reject HO, with corresponding correct decisions and errors.
P-values and significance:
P-value indicates the probability of obtaining the observed data (or more extreme) if HO is true.
If p ≤ α (commonly α = 0.05), reject HO and report statistical significance.
Cautions:
P-values depend on the data and sample size;
They do not measure probability that HO is true; they reflect data compatibility with HO.
Multiple testing can inflate Type I error (problem of multiple comparisons).
Clinical relevance vs statistical significance:
A finding can be statistically significant but clinically trivial
Researchers should argue for clinical relevance despite p-values lacking a strict objective threshold.
Probability, Hypothesis Testing, and Power
Probability basics (example from transcript):
If a coin has equal chance of heads/tails, the probability of heads is 0.5 (50%).
Random error due to chance means observed results may deviate from exact probabilities in a finite sample.
Statistical significance and error thresholds:
Alpha level (α): the maximum probability of committing a Type I error researchers are willing to accept; commonly 0.05.
Significance decision rule: if the observed probability (ρ) of the result under HO is less than α, reject HO.
P-values and interpretation:
P-values are not the probability that HO is true; they are conditional on HO being true.
They provide a measure of how surprising the observed data would be if HO were true.
Hypotheses, Population Parameters, and Sampling
Key terminology:
Population: large group with defined characteristics.
Sample: subset of participants drawn from the population.
Parameter: a value that describes a characteristic of a population (e.g., population mean μ).
Statistic: a value that describes a characteristic of a sample (e.g., sample mean
ar{X}).
Sampling principles:
Sample should be large enough to be representative, yet small enough to be practical.
Often chosen to achieve a desired statistical power (commonly 0.8).
Power and sample size planning:
Power reflects the probability of correctly rejecting HO when it is false.
Higher power requires larger samples or larger effect sizes; commonly a power of 0.8 is targeted.
Recap on inference workflow:
State HO and HA.
Choose a test statistic and significance level α.
Compute p-value or test statistic.
Compare to critical value or use p-value to make a decision.
Report CI and effect size to discuss practical significance.
Worked Examples and Practical Notes
Example 1 (Central Tendency): Data = {1, 5, 4, 3, 2}
Mean = ar{X} = rac{1+5+4+3+2}{5} = 3
Population mean (if applicable) denoted by mu; sample mean is ar{X}.
Example 2 (Range and IQR): Data set (ordered) = {2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10}
Range = 10 - 2 = 8
Q1 = 4, Q3 = 7 → IQR = Q3 - Q1 = 3
Example 3 (Normal curve rules):
68% within ±1 SD, 95% within ±2 SD, 99% within ±3 SD.
Example 4 (Confidence Interval):
If sample mean is ar{X}, SE = SE = rac{s}{ sqrt{n}}, then a 95% CI is ar{X} z_{0.025} imes SE = ar{X} 1.96 imes SE.
Example 5 (Power planning):
To achieve power around 0.8, design the study with a sufficient sample size given an expected effect size and variability.
Practical takeaways:
Use mean for symmetric interval/ratio data; use median for skewed data orOrdinal data when outliers are present.
Report both a measure of central tendency and a measure of variability (e.g., mean ± SD or median and IQR).
When performing inferential statistics, clearly distinguish statistical significance from practical/clinical significance.
Notes and references in this set are based on the provided transcript content from the Basic Statistical Concepts chapter (Descriptive vs Inferential Statistics) and summarize the major concepts, formulas, and interpretations presented.