Describing Data with Tables and Graphs

Frequency Tables

Frequency tables are used to organize data into categories or intervals, showing how often each value or range of values occurs.

• Frequency: The number of times a value or category appears in the dataset.

• Relative Frequency: The proportion of the total number of data points that fall into each category.

• Cumulative Frequency: The running total of frequencies through the classes in ascending order.

• Purpose: Frequency tables help summarize large datasets and are the basis for constructing histograms and bar charts.

Example: A frequency table for test scores might show how many students scored within each 10-point interval.

Boxplots

Boxplots (box-and-whisker plots) visually summarize the distribution of a dataset, highlighting the median, quartiles, and potential outliers.

• Median: The middle value of the dataset.

• Quartiles: Values that divide the data into four equal parts (Q1, Q2/median, Q3).

• Whiskers: Lines extending from the box to the smallest and largest values within 1.5 times the interquartile range (IQR) from the quartiles.

• Outliers: Data points outside the whiskers, often marked with dots or asterisks.

Example: A boxplot can show the spread and symmetry of exam scores in a class.

Describing Data Numerically

Measures of Center

Measures of center describe the typical or central value in a dataset.

• Mean (Arithmetic Average): The sum of all data values divided by the number of values. Formula:

• Median: The middle value when data are ordered from least to greatest. If there is an even number of values, the median is the average of the two middle values.

• Mode: The value(s) that occur most frequently in the dataset. There can be more than one mode or none at all.

Example: For the data set {2, 4, 4, 5, 7}, the mean is 4.4, the median is 4, and the mode is 4.

Normal Distribution and Continuous Random Variables

Normal Distributions

The normal distribution is a symmetric, bell-shaped curve that describes many natural phenomena.

• Properties: Symmetrical about the mean, mean = median = mode, defined by mean (μ) and standard deviation (σ).

• Standard Normal Distribution: A normal distribution with mean 0 and standard deviation 1.

Example: Heights of adult humans are approximately normally distributed.

Finding Z-Scores

A z-score indicates how many standard deviations a data point is from the mean.

• Formula:

• Interpretation: A positive z-score means the value is above the mean; a negative z-score means it is below the mean.

Example: If a test score is 85, the mean is 80, and the standard deviation is 5, then .

Using the Standard Normal Table to Find Percentiles

The standard normal table (z-table) provides the probability that a standard normal variable is less than a given z-score.

• Percentile: The percentage of data below a particular value.

• Procedure: Find the z-score, then use the table to determine the corresponding percentile.

Example: A z-score of 1.0 corresponds to the 84th percentile (approximately 84% of values are below this score).

Probability

Basic Probability Concepts

Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).

• Experiment: A process that leads to one of several possible outcomes.

• Sample Space (S): The set of all possible outcomes.

• Event: A subset of the sample space.

• Probability of an Event:

Example: The probability of drawing a heart from a standard deck of cards is .

Probability Questions: Cards, Marbles, Dice

• Deck of Cards: 52 cards, 4 suits, 13 ranks per suit.

• Bag of Marbles: Probability depends on the number and color of marbles.

• Dice: A standard die has 6 faces, each equally likely.

Example: Probability of rolling a 4 on a die: .

Intro to Stats and Collecting Data

Converting Units, Fractions, Decimals, and Percentages

Converting between different representations of numbers is essential for data analysis and interpretation.

• Fractions to Decimals: Divide the numerator by the denominator.

• Decimals to Percentages: Multiply by 100 and add the % symbol.

• Percentages to Fractions: Write the percentage over 100 and simplify.

• Unit Conversion: Use conversion factors to change from one unit to another (e.g., inches to centimeters).

Example: 0.75 = 75% = 3/4.

Converting Scientific Notation

Scientific notation expresses very large or small numbers in the form .

• To Decimal: Move the decimal point n places to the right (if n is positive) or left (if n is negative).

• To Scientific Notation: Place one nonzero digit to the left of the decimal, count the decimal places moved as the exponent.

Example: .

Finding Absolute and Relative Change

These measures describe how much a quantity has increased or decreased, both in actual and proportional terms.

• Absolute Change: The difference between the new and original values. Formula:

• Relative Change: The absolute change divided by the original value, often expressed as a percentage. Formula:

Example: If a price increases from $50 to $60, absolute change is $10, relative change is $10/$50 = 0.2 or 20%.

Finance Applications in Statistics

Simple Interest

Simple interest is calculated only on the original principal amount.

• Formula: where I = interest, P = principal, r = annual interest rate (decimal), t = time in years.

Example: I = 1000 \times 0.05 \times 2 = $100$.

Compounding Interest

Compound interest is calculated on the principal and on previously earned interest.

• Formula: where A = final amount, P = principal, r = annual interest rate, n = number of compounding periods per year, t = years.

Example: A = 1000 \times (1 + 0.05)^2 = .

Savings and Loans

• Savings: Regular deposits into an account can be modeled using the future value of an annuity formula.

• Loans: Loan payments can be calculated using the present value of an annuity formula.

Example: Saving $100 per month at 6% annual interest compounded monthly for 5 years.

Linear and Exponential Growth Models

Linear Growth

Linear growth increases by the same amount in each time period.

• Formula: where m = rate of change, b = initial value.

Example: A population increases by 100 people each year.

Exponential Growth and Population Doubling

Exponential growth increases by the same percentage (rate) in each time period.

• Formula: where a = initial amount, b = growth factor, x = number of periods.

• Doubling Time: The time it takes for a quantity to double in size. Formula:

Example: A population of 1000 grows by 5% per year: .

Summary Table: Key Statistical Measures