Introduction to Statistics

Introduction to Statistics

• Statistics is the process of collecting, organizing, analyzing, interpreting, and presenting data.

Types of Statistics

• Descriptive Statistics: Used to describe and summarize data. It uses measures of central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation) to explain data characteristics.

• Inferential Statistics: Used to draw conclusions and make predictions from data. It involves statistical tests on samples to make inferences about a larger population.

Types of Data

• Qualitative Data: Descriptive or categorical data that cannot be measured numerically, such as gender, color, or location.

  • Nominal Data: Labels variables without any quantitative value (e.g., male/female, hair color). Examples include:

    • Nationality (American, German, Filipino)
    • Color (Blonde, Black, Brown)

  • Ordinal Data: Data with a natural order (e.g., satisfaction levels, rankings). Examples include:

    • Very likely, Likely, Neutral, Unlikely, Very unlikely

• Quantitative Data: Numerical data that can be measured or counted.

  • Discrete Data: Consists of integers or whole numbers; countable and finite (e.g., number of students in a class). Examples include:

    • Total numbers of students present in a class
    • Cost of a cell phone
    • Numbers of employees in a company
    • The total number of players who participated in a competition
    • Days in a week

  • Continuous Data: Fractional numbers; can be divided into smaller levels (e.g., height, weight). Examples include:

    • Height of a person
    • Speed of a vehicle
    • "Time-taken" to finish the work

• Interval Data: Data with equal intervals between each point. Examples include: - IQ Test - NAT Test - Age - Temperature, in degrees Fahrenheit or Celsius

• Ratio Data: Quantitative data with a true zero point and equal intervals between points. - Weight, height, length - Temperature in Kelvin - Area

Measures of Central Tendency

• Mean: The sum of all observations divided by the total number of observations.

• Median: The middle value in an ordered set.

• Mode: The most frequently occurring value in a data set.

Measures of Relative Position

• Measures of position describe the position of a single value relative to other values in a set

• Quartiles: Values that divide a dataset into four equal parts (Q1, Q2, Q3).

• Percentiles: Values that divide a dataset into 100 equal parts.

• Deciles Divide a distribution into ten equal parts. The formula is Dk = k(n+1)/10 where k = the desired decile (1 to 10) and n = the number of observations

Measures of Dispersion

• Dispersion in statistics describes how spread out a set of data is. Range, variance, and standard deviations are main measures of dispersion

• Range: The difference between the largest and smallest value in the data.

• Variance: A measure of how spread out the data is from the mean; calculated by summing the squared differences between each data point and the mean, divided by the total number of data points.

  • Population variance: σ^2 = Σ(x - μ)^2 / N

Random Variables

• Random variable: Used to quantify the outcome of a random experiment. Possible values are determined by chance.

  • Discrete Random Variable: Can take on a finite number of distinct values

    • Example: Number of children in a family

  • Continuous Random Variable: Can take on an infinite number of values

    • Example: Weight of a person

Probability Mass Function

• Probability mass function defines the probability that a discrete random variable will be exactly equal to a particular value

• The probability of each value is between 0 and 1 inclusive. Σf(x)=1; sum of all probabilities is 1

Normal Distribution

• A bell-shaped, symmetrical distribution where the mean, median, and mode are equal. Deviations from this can give positive or negative skew

Z-score

• A standard score representing the number of standard deviations a raw score is above or below the mean.

  • Z = (x - μ) / σ