Bound reference

Categorical Data

  • Definition: Variables representing qualities or characteristics of groups or categories.

    • Examples: Hair color, gender, ratings (high, medium, low; or 1, 2, 3)

Types of Data

Numerical Data

  • Definition: Variables that represent quantities or measurements that are countable.

    • Examples: Height, mass, salaries, number of seats on a bus

Categorical Data

  • Nominal Data: Categorical data without an associated order or scale.

    • Examples: Hair color, method of transport to school

  • Ordinal Data: Categorical data aligned to a qualitative scale.

    • Examples: Rating scale (strongly disagree to strongly agree)

Discrete Data

  • Definition: Can take only particular fixed values, usually whole numbers.

    • Examples: The number of beads on an abacus

    • Discrete data can be counted.

Continuous Data

  • Definition: Can take all values, usually within a specific range.

    • Examples: The height of a child

    • Continuous data can be measured.

Population and Sample

Population

  • Definition: Entire category or group of interest.

    • Census: Study of everyone or everything in a population.

Sample

  • Definition: A small group from the population.

    • Importance: Should be representative and have sufficient participants to portray variety.

  • Sampling Bias: Occurs when the data collection method produces a sample that does not accurately reflect the population.

Sampling Methods

Convenience Sampling

  • Description: Selecting participants who are easily obtainable.

    • Example: Approaching individuals outside a shopping center.

  • Benefits: Quick, easy, less expensive.

  • Negatives: May not be representative of the population; may be biased.

Random Sampling (Simple Random Sampling)

  • Description: Participants are chosen from the population, each having an equal chance of selection.

    • Example: Using a random name generator or picking names from a hat.

  • Benefits: Usually representative of the population; reduces sampling bias.

  • Negatives: Time-consuming.

Systematic Sampling

  • Description: Selecting participants at regular intervals.

    • Example: Numbering a list and picking every 5th then every 8th person.

  • Benefits: Usually representative of the population; reduces sampling bias.

  • Negatives: Time-consuming.

Stratified Sampling

  • Definition: Sampling using proportional representation.

    • The population is divided into strata, with sample size proportional to size of each subgroup.

    • Example: Surveying a favorite soap opera among different age categories.

    • Steps to conduct:

      • Identify population and strata.

      • Randomly select samples from each strata.

Comparing Data

Measures of Center

  • Definition: Statistics that describe a data set by identifying its center.

    • Three Measures: Mean, median, mode.

Measures of Spread

  • Definition: Describe how similar or varied the data values are.

    • Includes: Range, interquartile range, standard deviation.

Measures of Central Tendency

Mean

  • Definition: The average of a data set.

  • How to Calculate: Sum of all values divided by the number of values.

  • When Not to Use: If there are outliers or the data is skewed.

Median

  • Definition: The middle value of an ordered data set.

  • How to Calculate: Order values, find middle number. If two middle values, average them.

Mode

  • Definition: The value that occurs most frequently.

    • Bimodal: Two values occur most frequently.

    • Multimodal: Three or more values occur at maximum frequency.

    • No Mode: When there are no repeating values.

Measures of Spread

Range

  • Definition: Difference between max value and min value.

Interquartile Range (IQR)

  • Definition: Difference between upper (Q3) and lower quartile (Q1).

  • IQR Formula: IQR = Q3 - Q1.

Standard Deviation

  • Definition: Measure of how far each data value is from the mean.

    • Larger standard deviation indicates more spread out values; lower indicates consistency.

Finding Quartiles

Steps to Determine Quartiles

  1. Order the data and find the median (Q2).

  2. Identify lower half and find its median (Q1).

  3. Identify upper half and find its median (Q3).

Outliers

Steps to Identify Outliers

  1. Sort data from low to high.

  2. Identify Q1, Q2, and Q3.

  3. Calculate IQR = Q3 - Q1.

  4. Calculate lower fence: Q1 - (1.5 * IQR).

  5. Calculate upper fence: Q3 + (1.5 * IQR).

  6. Identify values outside these fences as outliers.

Box Plots

  • Definition: Graphical representation of a five-number summary.

    • Includes minimum, Q1, median (Q2), Q3, and maximum.

  • Construction: Box represents the interquartile range (middle 50% of data), whiskers extend to min and max excluding outliers.

Shape of Data Distribution

Symmetrical Data Distribution

  • Definition: A distribution with a central line of symmetry.

    • Mean, median, and mode are equal.

Skewed Data Distributions

  • Positively Skewed: Data is concentrated in the lower part.

  • Negatively Skewed: Data is concentrated in the upper part.

Implications for Measures of Center and Spread

  • Use median and IQR for skewed distributions or those with outliers.

  • Use mean and standard deviation for symmetrical data.

Normal Distribution

  • In symmetrical bell-shaped data:

    • Approximately 68% of data lies within one standard deviation from the mean.

    • Approximately 95% lies within two standard deviations.

    • Approximately 99.7% lies within three standard deviations.