Intro to Statistics

Statistical and Critical Thinking

Statistics is the science of collecting, analyzing, interpreting, and drawing conclusions from data. It is essential for making informed decisions in the presence of variability and uncertainty.

• Definition of Statistics: The science of planning studies and experiments, obtaining data, organizing, summarizing, presenting, analyzing, and interpreting those data, and drawing conclusions based on them.

• Scope in Course: This course focuses primarily on analyzing and interpreting data, rather than on planning studies or organizing data.

What is Data?

Data refers to any recorded observation, whether numerical or categorical. It can include scientific measurements (such as distances, weights, or temperatures) as well as non-numerical information (such as survey responses or categories).

• Data: Any observation that can be recorded and categorized in some way.

• Examples: Heights of students, favorite colors, survey responses.

Populations, Samples, and Census

Populations and Census

In statistics, we often aim to make inferences about a population—the entire group of individuals or items of interest. A census involves collecting data from every member of the population, which is often impractical for large populations.

• Population: The complete set of individuals or items being studied (e.g., all students at a university).

• Census: Data collection from every member of the population.

• Example: Surveying every business in Japan to determine average revenue.

Samples

Because censuses are often impractical, we typically collect data from a sample, a subset of the population. Conclusions about the population are then drawn from the sample data.

• Sample: A subset of the population used to make inferences about the whole.

• Example: Measuring the heights of 6 students at a university to estimate the average height of all students.

Problems with Samples

Not all samples are equally reliable. Several issues can affect the validity of sample-based conclusions:

• Voluntary Response Samples: Individuals select themselves to participate, often leading to bias (e.g., online polls, product reviews).

• Non-Random Sampling: If the sample is not randomly selected, it may not represent the population accurately.

• Sample Size: Small samples are more likely to be unrepresentative due to random chance.

• Self-Reporting Bias: Responses may be inaccurate due to the way questions are asked or due to respondents' motivations.

Example: Online product reviews often have more extreme ratings (1 or 5 stars) because people are more motivated to review when they have strong opinions.

Statistical vs. Practical Significance

It is important to distinguish between results that are statistically significant and those that are practically significant:

• Statistical Significance: When the observed effect in a sample is unlikely to have occurred by random chance (commonly, if the probability of coincidence is less than 5%).

• Practical Significance: Whether the observed effect is large enough to be meaningful in a real-world context.

Example: If 99 out of 100 people like a product, this is statistically significant. If a coin lands heads 54 out of 100 times, this is likely due to chance and not statistically significant.

Types of Data

Major Categories of Data

Data can be classified into two broad categories:

• Qualitative (Categorical) Data: Describes qualities or categories (e.g., colors, brands, yes/no responses).

• Quantitative (Numerical) Data: Consists of numbers representing counts or measurements (e.g., height, weight, age).

Types of Quantitative Data

• Discrete Data: Countable values (e.g., number of students in a class).

• Continuous Data: Measurable quantities that can take any value within a range (e.g., height, temperature).

Levels of Measurement

Data can also be classified by the level of measurement, which determines the types of statistical analyses that are appropriate.

• Nominal: Data are labels or names with no inherent order (e.g., gender, colors).

• Ordinal: Data have a meaningful order, but differences between values are not meaningful (e.g., rankings, letter grades).

• Interval: Differences between values are meaningful, but there is no true zero (e.g., temperature in Celsius or Fahrenheit, years).

• Ratio: Like interval, but with a true zero point, allowing for ratios (e.g., height, weight, age, temperature in Kelvin).

Parameters, Statistics, and Missing Data

Parameter vs. Statistic

• Parameter: A numerical summary describing a characteristic of a population (e.g., population mean \( \mu \)).

• Statistic: A numerical summary describing a characteristic of a sample (e.g., sample mean \( \bar{x} \)).

Example: The average height of all students at a university is a parameter; the average height of a sample of students is a statistic.

Missing Data

Sometimes, data values are missing. The reason for missing data affects how we handle it:

• Missing Completely at Random (MCAR): The missingness is unrelated to the data itself (e.g., data lost due to a technical error).

• Missing Not at Random (MNAR): The missingness is related to the value or reason it is missing (e.g., people not answering sensitive survey questions).

Handling Missing Data

• Deletion: Remove cases with missing data. Appropriate if data are missing completely at random.

• Imputation: Substitute missing values with estimated values (e.g., mean, median, or randomly generated values). The method should be chosen based on the context and reason for missingness.

Example: If survey responses are missing at random, deleting those cases may be acceptable. If not, imputation or more advanced methods may be necessary to avoid bias.