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Statistics

The branch of mathematics involving the collection, organization, analysis, interpretation, and presentation of data.

Population

The entire collection of individuals or items of interest in a statistical study.

Sample

A subset of individuals or items selected from a larger population.

Parameter

A numerical measurement describing a characteristic of an entire population.

Statistic

A numerical measurement describing a characteristic of a sample.

Descriptive Statistics

Methods focused on organizing, summarizing, and presenting data through tables, graphs, and numerical summaries.

Inferential Statistics

Methods that use sample data to draw conclusions, estimations, or predictions about a population.

Qualitative Data

Categorical information described by non-numerical characteristics, attributes, or labels.

Quantitative Data

Numerical information resulting from counts or measurements.

Discrete Data

Quantitative values that are countable, with distinct gaps between possible numbers.

Continuous Data

Quantitative values that can take on infinitely many possible numerical values along a continuous scale.

Population Mean Formula

$$\mu = \frac{\sum x}{N}$$

Sample Mean Formula

$$\bar{x} = \frac{\sum x}{n}$$

Median

The middle observation in a dataset when numbers are arranged in ascending or descending order.

Mode

The value or values that appear with the highest frequency in a given dataset.

Range

The difference between the maximum value and the minimum value in a dataset.

Standard Deviation

A measure of variation that quantifies how much data values deviate from the mean.

Statistics

The branch of mathematics involving the collection, organization, analysis, interpretation, and presentation of data. Example: Analyzing survey responses from $1000$ citizens to predict election results.

Population

The entire collection of individuals or items of interest in a statistical study. Example: All registered voters in a country.

Sample

A subset of individuals or items selected from a larger population. Example: $500$ registered voters selected to participate in an exit poll.

Parameter

A numerical measurement describing a characteristic of an entire population. Example: The actual average age of all citizens in a country, calculated using census data.

Statistic

A numerical measurement describing a characteristic of a sample. Example: The average age calculated from a survey of $100$ citizens.

Descriptive Statistics

Methods focused on organizing, summarizing, and presenting data through tables, graphs, and numerical summaries. Example: Creating a bar chart to display monthly sales.

Inferential Statistics

Methods that use sample data to draw conclusions, estimations, or predictions about a population. Example: Using a sample of $100$ light bulbs to estimate the average lifespan of all light bulbs produced by a factory.

Qualitative Data

Categorical information described by non-numerical characteristics, attributes, or labels. Example: Eye colors, hair colors, or favorite soda brands.

Quantitative Data

Numerical information resulting from counts or measurements. Example: Heights of students, daily temperature, or number of cars in a parking lot.

Discrete Data

Quantitative values that are countable, with distinct gaps between possible numbers. Example: The number of students in a classroom ($25$, $26$, but not $25.5$).

Continuous Data

Quantitative values that can take on infinitely many possible numerical values along a continuous scale. Example: The exact time taken to run a race ($12.34\,s$).

Population Mean Formula

$\mu = \frac{\sum x}{N}$\n\nExample: If a population consists of values $2$, $4$, and $6$, then $\mu = \frac{2 + 4 + 6}{3} = 4$.

Sample Mean Formula

$\bar{x} = \frac{\sum x}{n}$\n\nExample: If a sample of test scores is $80$, $90$, and $100$, then $\bar{x} = \frac{80 + 90 + 100}{3} = 90$.

Median

The middle observation in a dataset when numbers are arranged in ascending or descending order. Example: For the ordered dataset $3$, $5$, $7$, $9$, $11$, the middle value is $7$.

Mode

The value or values that appear with the highest frequency in a given dataset. Example: In the dataset $2$, $3$, $3$, $4$, $5$, the most frequent value is $3$.

Range

The difference between the maximum value and the minimum value in a dataset. Example: For data values $4$, $8$, $15$, $16$, $23$, $42$, the difference is $42 - 4 = 38$.

Standard Deviation

A measure of variation that quantifies how much data values deviate from the mean. Example: A value of $2\,cm$ in plant heights indicates that most plants are close to the average height, whereas a value of $15\,cm$ indicates wide variation.

Nominal Level of Measurement

Data that consists of categories, names, or labels only, which cannot be arranged in an ordering scheme. Example: Survey responses like Yes, No, or Undecided, or types of pet (Dog, Cat, Fish).

Ordinal Level of Measurement

Data that can be arranged in some order, but differences between data values either cannot be determined or are meaningless. Example: Course grades (A, B, C, D, F) or star ratings ($1$ star to $5$ stars).

Interval Level of Measurement

Data that can be ordered, and meaningful differences between values can be calculated, but there is no natural zero starting point. Example: Temperatures in Fahrenheit ($0^\circ F$ does not mean complete absence of heat).

Ratio Level of Measurement

Data that can be ordered, with meaningful differences and a true natural zero point where zero indicates the absence of the quantity. Example: Distance traveled ($0\,km$ means no distance traveled, and $10\,km$ is twice as far as $5\,km$).

Simple Random Sample

A sampling method where every sample of size $n$ has an equal chance of being selected from the population. Example: Drawing names out of a hat or using a computer random number generator to select $20$ employees.

Stratified Sampling

A sampling method where the population is divided into subgroups (strata) sharing similar characteristics, and a sample is drawn from each subgroup. Example: Grouping students by grade level ($9\text{th}$, $10\text{th}$, $11\text{th}$, $12\text{th}$) and randomly selecting $25$ students from each grade.

Cluster Sampling

A sampling method where the population is divided into sections (clusters), whole clusters are randomly selected, and all members from those selected clusters are surveyed. Example: Randomly choosing $5$ specific school districts in a state and surveying every teacher in those $5$ districts.

Variance

A measure of variability equal to the square of the standard deviation. Example: If the standard deviation of test scores is $5$, this value equals $5^2 = 25$.