Statistics Basics Quiz

Mathematics of Data Management

• Part 1 covers Descriptive Statistics, making up 30% of the course.
• Course notes are from Lower Canada College.

Unit 1 - Introduction to Statistics

• A. Basics
  • A survey is a process of gathering information for informed decisions.
  • Data are observations (e.g., eye color, salary, height).
  • A population is the complete group.
  • A sample is a subset of the population.
  • Sampling Techniques
    • Voluntary Response Sample: Participants decide whether to participate. This method has limitations, as the sample may not represent the population
    • Simple Random Sample: Participants are selected randomly. This approach can ensure the sample represents the population fairly. An example is selecting 10 students from each grade randomly.
  • Sources of Bias
    • Sampling Bias: When the sample doesn't reflect the characteristics of the whole population. An example is only surveying Montreal Canadiens fans to determine favourite NHL team.
    • Non-Response Bias: When specific groups aren't represented in a survey because they opted out of participating. This often arises when surveys are optional. An example is when only 50% of students respond to a survey on athletics.

Generating A Simple Random Sample

• Steps to generate a simple random sample using a calculator are described, including seeding the random number generator.

Types of Data

• Qualitative Data: Characterized by names or labels. Examples are eye color, political party affiliations.
• Quantitative Data: Characterized by numerical measurements.
  • Discrete Data: Finite or countable values. Examples include the number of eggs laid in a week, rolls of a die.
  • Continuous Data: Infinite possible values. An example is the amount of milk a cow yields in a year (any value between 0 to 7000 liters).

Levels of Measurement

• Nominal: Categories, no order (e.g., favorite food).
• Ordinal: Categories with an order (e.g., letter grade).
• Interval: Has meaningful differences between values but no true zero (e.g., temperature in Celsius).
• Ratio: Has meaningful differences and a true zero (e.g., salary, age).

C. Collecting Data

• Sampling Techniques
  • Systematic Sampling: Select participants at regular intervals. Good for large populations, but potentially prone to bias if the intervals have a hidden pattern
  • Stratified Sampling: Divides the population into strata (groups with shared characteristics). Random sampling within each strata. Can improve representation, but also requires significant effort
  • Cluster Sampling: Divide the population into clusters, randomly select clusters. Can be more efficient with large populations, but may reduce diversity.
  • Convenience Sampling: Selecting accessible participants. This method usually isn't reliable, as the sample is unlikely to represent the population truly.

Unit 2 - Graphing and Summarizing Data

• A. Graphing data
  • Data visualisation facilitates easier understanding and makes predictions.
  • Creating bar charts, pie charts, and Pareto charts is discussed.
  • Graphs organise and summarise data, allowing quicker analysis.
• Frequency Distribution Table
  • Tabulates data with frequencies, relative frequencies and cumulative frequencies
• Histograms
  • Bars are side by side, similar to bar graphs, where each bar shows frequency of data within a class interval
• Frequency Polygon
  • Connect the midpoints of adjacent bars
• Ogive (Cumulative Frequency Polygon) -Shows cumulative frequencies
• Stem and Leaf Diagrams -Visualises data by separating each data value into a stem and leaf

Measures of Central Tendency

• Mean: Average of the data values
• Median: Middle value when data is ordered
• Mode: Most frequent value

Measures of Dispersion (Spread)

• Range: Difference between highest and lowest data values.
• Variance: Average of the squared differences from the mean.
• Standard Deviation: Square root of the variance.

Finding the mean, variance and standard deviation

• Steps for calculating mean, variance and standard deviation are illustrated with an example of dog heights.
  • Calculate the mean (average) of the data values
  • Calculate the difference between each value and the mean, and square these differences
  • Calculate the average (mean) of the squared differences
  • Calculate the square root of the variance to get the standard deviation

Unit 3 - Probability

• A. Basics of Probability
  • Probability is about the likelihood of an event occurring. An example of an event is getting a boy or a girl.
  • An event is a group of outcomes from a particular procedure.
  • A simple event has no subsets. The sample space is the whole group of all the possible simple events.
  • Probability of an event is between zero and one.
• Types of Events
• Complementary Events: The events that do not occur. The complement of event A occurs if A doesn't occur
• Compound Events A compound event occurs if two or more simple events occur. An example is if both A and B occur
• Independent/Dependent Events: If the occurence of event one does not affect the probability of the other (indepedent), or if the occurrence of one event does affect the probability of the other (dependent).
• Conditional probability The probability of an event given some additional information, that some other event has already occurred.

Counting

• Permutations: Order matters.
• Combinations: Order does not matter.
• Rules for calculating them are given, for both when all items are different, or some items are the same.
  • Factorial Rule: Calculating permutations when there are the same number of items as options
  • Fundamental Counting Rule: Calculating possible outcomes of multiple events
  • Permutation Rule: Calculating permutations when there are multiple items that are identical

Unit 4 - The Normal Distribution

• A. Normal Distributions and standard deviations:

• Normal curves are symmetrical and bell shaped

• mean, median, mode are equal and centered in the distribution

• 68% of data values are within one standard deviation of the mean

• 95% are within two standard deviations of the mean

• 99.7% are within three standard deviations of the mean. -z-scores convert any normal distribution to a standard normal distribution. Standardized z-scores allow for comparisons between different distributions.

• Range Rule of Thumb: The vast majority of values live within 2 standard deviations of the mean

• B. Skewness:

  • Visual representation of how the distribution is lopsided or not
  • Measures of skewness quantify how symmetrical the distribution is

• C. Standard Normal distributions and z-scores:

  • Allows for comparison between different distributions

• E. Percentages and values (Normal Distribution):

  • Identifying specific values (e.g., heights) that fall within given percentiles of a normal distribution.

• F. Proving Normalcy:

  • Identify if distribution satisfies characteristics of a normal distribution