Introduction to statistics

• Statistical Methods I is MATH 153 and is taught by Collins Abaitey.

Course Outline

• The course covers Introduction to Statistics, Frequency Distributions and Graphs, Measures of Central Tendency, Measures of Variation, Measures of Position, Probability and Counting Rules, Random Variables, and Discrete Probability Distributions.
• A recommended textbook for the course is Elementary Statistics (A Step by Step Approach) by Allan G. Bluman.

Introduction to Statistics

• Data consists of information from observations, counts, measurements, or responses.
• Statistics is the science of collecting, organizing, analyzing, and interpreting data to make decisions.
• A population includes all outcomes, responses, measurements, or counts of interest.
• A sample is a subset of a population.
• The population is the responses of all adults in Ghana when surveying 2500 adults in Ghana on global warming.
• The sample includes the responses of the 2500 adults in Ghana surveyed.
• A parameter describes a population characteristic, like the average age of all people.
• A statistic describes a sample characteristic, like the average age of a sample of people.
• The average starting salary for petroleum engineers of $83,121 represents a statistic, as it is based on a sample.
• The average cut-off point of aggregate 12 for the 2,182 students admitted to KNUST in 2009 is a parameter since it is derived from the entire population of admitted students.
• Descriptive Statistics involves organizing, summarizing, and displaying data.
• Inferential Statistics involves using a sample to draw a conclusion about a population.
• In a study of men aged 48 over 18 years, descriptive statistics include statements like "70% of unmarried men were alive at age 65" and "90% of married men were alive at age 65.".
• A possible inference from the men study is that being married is associated with a longer life for men.

Data Classification

• Qualitative data consists of attributes, labels, or non-numerical entries like place of birth, eye color, or political affiliation.
• Quantitative data consists of numerical measurements or counts, such as age, weight, or temperature.
• Nominal level measurements are qualitative and cannot be ordered.
• Ordinal level measurements are qualitative or quantitative, can be ordered, but differences between data entries are not meaningful.
• Top five TV programs from 5/4/09 to 5/10/09 (American Idol, Dancing with the Stars, NCIS, The Mentalist) are ranked on an ordinal level.
• Network Affiliates in Pittsburgh, PA (WTAE, WPXT, KDKA and WPGH) are an example of a nominal level.
• Interval level measurements are quantitative, can be ordered, have meaningful differences between data entries, and zero represents a position on a scale, but a ratio doesn't make sense.
• Ratio level measurements are similar to interval, but zero is inherent and implies 'none,’ which allows for meaningful ratios.
• The New York Yankees' World Series victories (years) are measured on an interval level due to the meaningful differences between the years.
• 2009 American League Home Run Totals are measured on a ratio level because a ratio of the home runs can be calculated and is meaningful.

Data Collection and Sampling Techniques

• Data collection techniques include observational studies, experiments, simulations, and surveys.
• Sampling techniques include simple random sampling, stratified sampling, cluster sampling, and systematic sampling.

Frequency Distribution and Graphs

• A frequency distribution organizes raw data into a table using classes and frequencies.
• Class width = 10, calculated by 11 - 1 = 10.
• class mark = 5.5, calculated by (1 + 10) / 2
• The lower class limit for the class 1–10 is 1.
• The upper class limit for the class 1-10 is 10.
• The lower class boundary for the class 1 – 10 is 0.5.
• The upper class boundary for the class 1 – 10 is 10.5.
• class boundary = 0.5, by subtracting and adding 0.5 to the lower and upper class limits respectively.

Measures of Central Tendency

• Mean is the average found by summing all data entries and dividing by the number of entries, represented as μ = Σx / N for population mean and x = Σx / n for sample mean.
• Median is the middle value of an ordered data set.
• Mode is the most frequently occurring data entry.
• When ordering the data {200, 300, 400, 400, 500, 600, 700}, the mean is 442.9, the median is 400, and the mode is 400.
• When ordering the data {388, 397, 397, 427, 782, 872}, the mean is 543.8, the median is 412, and the mode is 397.
• For the data {100, 101, 102, 103, 104, 105, 106}, the mean and median are both 103, and there's no mode.
• For the data {250, 300, 300, 350, 350, 400, 450, 2000}, the mean is 550, the median is 350 and the modes are 300 and 350, making the set bimodal.
• Outliers greatly affect mean.
• Mean takes into account every entry of a data set.

Mean of Grouped Data

• In grouped data, the mean is calculated using x = Σfx / Σf.

Weighted Mean

• Weighted mean is calculated using x = Σwx / Σw, where w is the weight of each entry x.

Shapes of Frequency Distributions

• In a symmetric distribution, mean = median = mode.
• In a left-skewed distribution, mean < median < mode.
• In a right-skewed distribution, mode < median < mean.

Measures of Variation

• Range is the difference between the maximum and minimum data entries in a set.
• Deviation is the difference between a data entry, x, and the mean of the data set (x - μ or x - x̄).
• Population variance is σ² = Σ(x-μ)² / N.
• Sample variance is s² = Σ(x-x̄)² / (n-1).
• Population standard deviation is σ = √[Σ(x-μ)² / N].
• Sample standard deviation is s = √[Σ(x-x̄)² / (n-1)].
• For the data set {111, 112, 115, 117, 118, 119, 120}, the range is 9.

Measures of Position

• Quartiles divide an ordered data set into four approximately equal parts (Q1, Q2, Q3). -Deciles divide an ordered data set into ten equal parts (D1, D2 ... D9).
• Percentiles divide an ordered data set into 100 equal parts (P1, P2 ... P99).
• Range is calculated through IQR = Q3 - Q1.

Box Plots

• Box plots require a five-number summary: minimum entry, first quartile (Q1), second quartile/median (Q2), third quartile (Q3), and maximum entry.

Probability

• Probability is the chance of an event occurring.
• Probability experiment is a process that leads to well-defined outcomes.
• Outcome is the result of a single trial in a probability experiment.
• Sample space is the set of all possible outcomes.
• Event is a subset of the sample space.
• For probability experiment, a die roll has a sample space of {1, 2, 3, 4, 5, 6} .
• Classical probability is when outcomes in the sample space are equally likely, P(E) = n(E) / n(S).
• Empirical probability is the relative frequency of an event, P(E) = f / n.
• Subjective probability is based on intuition, guesses, and estimates.
• In rolling a six-sided die: The probability of rolling a 3 (Event A) = 1/6 and probability of rolling a number less than 5 (Event C) = 2/3.
• The probability of an event E falls between 0 and 1.
• Complement of event E is the set that includes the same sample space but not event E, denoted as E′.
• The complement of event E is equal to P(E') = 1 − P(E).

Conditional Probability and the Multiplication Rule

• It determines the probability of event B occurring when it is known that A has already occurred, P(B/A) = P(B∩A) / P(A).
• Events are independent if the occurrence of one does not affect the probability of the other and P(B/A) = P(B) or P(A/B) = P(A).
• For independent events, P(A and B) = P(A) * P(B).
• The outcome on a coin does not affect the probability of rolling a 6 on the die.

Mutually Exclusive Events and the Addition Rule

• Mutually exclusive events (A and B) cannot occur at same time, and P(A ∩ B) = 0.
• The probability that two events A or B will occur is described as P(A or B) = P(A ∪ B) = P(A) + P(B) − P(A ∩ B). However, if mutually exclusive, then P(A or B) = P(A ∪ B) = P(A) + P(B)
• An example in order to determine probabilities is with different types of blood that donors may have as they are mutually exclusive to each other.

Probability and Counting

• Counting Principles state if event M can occur in m number of times and event N can occur in n number of times, then the two events can occur in sequence is m * n.
• As an example, a scenario of looking at car manufacturers Ford, GM and Honda as the first event, then compact and midsize as the second event, then white(W), red(R), black(B), green(G) colour as the fourth sequence yields a result of 3 * 2 * 4 = 24 ways.
• Factorial Notation for any n number is done as n! = n * (n - 1) * (n - 2) ... 2 * 1. For example, 5! = 120.
• If there are many of the same element to account for such in a word it means that equation can be derived from doing n! / element1! + element2! ... elementn!.

Discrete Probability Distributions

• Random variables are variable whose values are determined by chance, generally referred to as X or Y.
• Has a countable amount of possible values. For example, the total chairs in a room.
• Can assume all values within an interval between any given values and can be measured. Is obtained for example, the temperature within 24 hours.
• Requirements for a probability distribution: 0≤𝑃(𝑋)≤1 and Σ𝑃(X)=1.