Unit 2: Introduction to Statistics and Data Analysis - Subjective Questions

Statistical inference is the process of using data collected from a sample to draw conclusions about the population from which the sample was selected.

It has two major components:

• Estimation: A sample statistic is used to estimate an unknown population parameter. For example, the sample mean estimates the population mean .
• Hypothesis testing: Sample evidence is used to assess a claim about a population parameter.

Because different samples can produce different results, statistical inference accounts for sampling variability. Reliable inference therefore requires a representative sample, an appropriate probability model, and a suitable method of analysis.

• A population is the complete collection of individuals or measurements of interest. Example: all students enrolled at a university.
• A sample is a subset selected from the population. Example: 200 students selected from that university.
• A parameter is a numerical characteristic of a population. Example: the mean height of all university students.
• A statistic is a numerical characteristic calculated from a sample. Example: the sample mean height of the 200 selected students.

Parameters are generally unknown, whereas statistics can be calculated from observed data and used to estimate parameters.

Random sampling gives every population member a known, and often equal, chance of selection. It helps create a representative sample and supports valid generalization from the sample to the population.

• Sampling bias is a systematic error caused by a sampling method that consistently favors certain outcomes or excludes parts of the population. A larger sample does not necessarily remove this bias.
• Sampling variability is the natural variation in statistics obtained from different random samples of the same population. It generally decreases as sample size increases.

Thus, random sampling primarily reduces selection bias, while a larger random sample generally reduces sampling variability.

The major principles are:

• Control: Keep extraneous conditions as similar as possible and use a control or comparison group when appropriate.
• Randomization: Randomly assign experimental units to treatments to balance known and unknown confounding factors.
• Replication: Apply treatments to several experimental units so that natural variability can be measured and treatment effects can be estimated more precisely.
• Blocking: Group similar experimental units into blocks and randomize treatments within each block. This reduces variation due to known nuisance factors.

Together, these principles help distinguish a genuine treatment effect from random variation and confounding.

In an observational study, researchers observe variables without assigning treatments or deliberately changing conditions. In a controlled experiment, researchers assign treatments to experimental units and compare responses under controlled conditions.

Causal inference is generally stronger in a randomized experiment because:

• Random assignment tends to balance confounding variables across treatment groups.
• A control group provides a basis for comparison.
• Experimental conditions can be standardized.
• The temporal order between treatment and response is clear.

An observational study can reveal association, but an unmeasured confounding variable may explain that association. Therefore, association alone does not establish causation.

For observations , the sample mean is the sum of all observations divided by the sample size:

For the given data, and

Therefore,

The sample mean is the arithmetic balance point of the data. It uses every observation and is consequently sensitive to extremely large or small values.

The sample median is the middle value of the ordered observations.

• If is odd, its position is .
• If is even, the median is the mean of the observations in positions and .

The data are already ordered and contain observations. The two middle observations are the third and fourth values, and . Thus,

The median divides the ordered data so that approximately half the observations lie on each side.

Both measures describe the center of a dataset, but they have different properties:

• The mean uses the numerical value of every observation and is useful for further algebraic and statistical analysis.
• The median depends primarily on the order of observations and is resistant to outliers.
• In a symmetric distribution without severe outliers, the mean and median are often close.
• In a right-skewed distribution, the mean is usually greater than the median; in a left-skewed distribution, it is usually smaller.

For a highly skewed distribution, the median is generally preferable because a small number of extreme observations can pull the mean toward the long tail, whereas the median remains relatively stable.

• Range: The difference between the maximum and minimum observations:
  
• Sample variance: The average squared deviation from the sample mean, with denominator :
  
• Sample standard deviation: The positive square root of the variance:
  
• Interquartile range: The spread of the middle of the ordered data:
  

The range is simple but highly sensitive to extremes. Variance and standard deviation use all observations, while the interquartile range is comparatively resistant to outliers.

Starting with the sum of squared deviations,

expand the square:

Therefore,

Since , this becomes

Using gives

Hence the computational form of sample variance is

First calculate the sample mean:

The squared deviations are

With , the sample variance is

The sample standard deviation is

Thus, observations typically lie about units from the mean, although standard deviation is not literally the average absolute distance. Variance is expressed in squared units, while standard deviation has the same units as the original data.

The sample variance is

Once the sample mean has been estimated, the deviations satisfy

Therefore, only deviations are free to vary; the last is determined by the others. This leaves degrees of freedom.

Dividing by would systematically underestimate the population variance . The correction from to , known as Bessel's correction, makes an unbiased estimator of under random sampling.

Discrete data take separate, countable values and are commonly obtained by counting. Examples include:

• Number of defective products in a batch
• Number of telephone calls received per hour
• Number of children in a household

Continuous data can theoretically take any value within an interval and are commonly obtained by measurement. Examples include:

• Height of a student
• Time required to complete a task
• Temperature of a chemical solution

A discrete variable often takes integer values, while a continuous variable may include decimals to the precision permitted by the measuring instrument.

A statistical model is a mathematical representation of how observed data are generated. It simplifies reality so that relationships can be described, predictions can be made, and uncertainty can be quantified.

A simple linear model is

where:

• is the response variable.
• is the deterministic component, representing the systematic relationship between and the mean response.
• is the random component, representing unobserved influences and measurement variation.

Common assumptions may include independent errors, mean-zero errors, constant error variance, and an appropriate error distribution. Conclusions from the model are reliable only when its assumptions are reasonably consistent with the data.

A systematic statistical investigation generally includes:

1. Define the problem: State the scientific question, objectives, population, and variables.
2. Formulate a hypothesis or model: Propose an explanation or relationship that can be examined using data.
3. Design the study: Select an observational or experimental design, sampling method, sample size, and measurement procedure.
4. Collect data: Follow a consistent protocol and record relevant contextual information.
5. Inspect and clean data: Check for missing values, impossible values, coding errors, and unusual observations without automatically deleting them.
6. Analyze data: Use numerical summaries, graphs, models, and inferential procedures.
7. Assess assumptions: Apply graphical diagnostics and other checks.
8. Interpret and communicate: Relate results to the original question, quantify uncertainty, discuss limitations, and report reproducible conclusions.

Scientific inspection is iterative because unexpected data patterns may require revisiting earlier stages.

Graphical diagnostics are plots used to assess whether data and model residuals support the assumptions of a fitted statistical model.

For residuals :

• A residual-versus-fitted plot checks linearity and constant variance. A random horizontal band around zero is desirable. Curvature suggests an incorrect functional form, while a funnel shape suggests nonconstant variance.
• A residual-versus-order plot can reveal trends, cycles, or dependence over time.
• A normal probability plot, or Q-Q plot, compares ordered residuals with theoretical normal quantiles. Approximate alignment along a straight line supports normality; systematic curvature or extreme departures may indicate skewness, heavy tails, or outliers.

Diagnostics do not prove that assumptions are true, but they can reveal important violations requiring model revision.

• A frequency distribution groups observations into classes and reports the count or relative frequency in each class. It provides a compact numerical summary.
• A histogram represents class frequencies with adjacent bars. It is effective for identifying shape, center, spread, skewness, modes, and possible gaps in moderate or large datasets.
• A stem-and-leaf display separates each observation into a stem and leaf. It shows distributional shape while preserving the original values, making it useful for small or moderate datasets.
• A dot plot places a dot for each observation on a number line. It clearly displays individual values, clusters, gaps, modes, and possible outliers in small datasets.

The appearance of a histogram depends on bin widths and boundaries, whereas stem-and-leaf displays and dot plots retain more information about individual observations.

A box plot is based on the five-number summary: minimum, first quartile , median, third quartile , and maximum, subject to the outlier rule.

Construction steps are:

1. Calculate , the median, and .
2. Calculate the interquartile range:
   
3. Determine the fences:
   
   
4. Draw a box from to with a line at the median.
5. Extend whiskers to the most extreme observations within the fences.
6. Plot observations beyond the fences individually as potential outliers.

The box length represents the middle of the data. Unequal whiskers or an off-center median may indicate skewness. Points beyond the fences require investigation but are not automatically errors.

A scatter plot represents each paired observation as a point in a two-dimensional coordinate system. It visually examines whether and how two quantitative variables vary together.

Important features include:

• Direction: An upward trend indicates positive association, while a downward trend indicates negative association.
• Form: The relationship may be linear, curved, clustered, or have another pattern.
• Strength: Points tightly grouped around a pattern indicate a stronger association.
• Outliers: Isolated points may strongly influence summaries or fitted models.
• Changing spread: A widening or narrowing vertical spread may indicate nonconstant variability.
• Subgroups: Separate clusters may reflect an additional categorical variable or different populations.

A scatter plot can reveal association, but by itself it cannot establish that changes in one variable cause changes in the other.

• Sample survey: Collects responses from a sample to describe a population. It is appropriate for estimating opinions, behaviors, or characteristics. Its limitations include nonresponse, response bias, and poor sampling frames.
• Observational study: Measures exposures and outcomes without treatment assignment. It is appropriate when experimentation is unethical or impractical. Its major limitation is possible confounding, which weakens causal conclusions.
• Controlled experiment: Deliberately assigns treatments, ideally through randomization. It is appropriate for estimating causal treatment effects. Limitations can include cost, ethical restrictions, noncompliance, and limited generalizability.
• Retrospective study: Uses past records or participants' recollections to compare previous exposures or characteristics. It is useful for studying rare outcomes or questions requiring long follow-up. Its limitations include incomplete records, selection bias, and recall bias.

The appropriate design depends on the research question, ethical constraints, available resources, and the type of conclusion required.