Untitled Quiz

• Statistical Methods of Applied Statistics are procedures and techniques used for data collection, presentation, analysis, and interpretation.

Descriptive Statistics

• Methods involve data collection, description, and analysis to describe a data set without drawing conclusions about a larger group.
• Focuses on clarifying obscure information within the given data.
• Conclusions are limited to the data being analyzed.

Inferential Statistics

• Methods involve making predictions or inferences about a larger data set, using information from a subset.

• Aims to predict and infer beyond mere description.

• Conclusions can be applied to a larger group, with the analyzed data considered a subset.

• Statistical Theory of Mathematical Statistics involves theoretical development which forms the basis of statistical methods.

Population and Sample

• Population: All elements under consideration in a study.
• Sample: A subset of the population from which data is collected.

Population and Sample Example

• A kerosene heater manufacturer wants to know if customers are satisfied.

• They contact 5,000 of their 200,000 customers.

• Population is all 200,000 customers.

• Sample is the 5,000 contacted customers.

• Parameter: A numerical characteristic of a population.

• Statistic: A numerical characteristic of a sample.

Parameter and Statistic Example

• A college polls 200 students and finds 12% smoke

• Population: all students at the college

• Sample: the 200 students polled

• Parameter: proportion of all college students who smoke

• Statistic: the 12% smoking rate in the sample

• Variable: A characteristic or attribute that can have different values for different entities.

• Measurement: Process of determining the value/label of a variable for an experimental or sampling unit.

• Experimental/Sampling Unit: The individual or object on which a variable is measured.

• Observation: A numerical recording of information on a variable.

• Data: A collection of observations.

Data Classification

• Primary Data: Original data measured by the researcher/agency that publishes it.
• Secondary Data: Data republished by another agency.
• Example: Philippine Statistics Authority data is primary, while subsequent publications are secondary.
• Internal Data: Relates to the organization's operations and functions.
• External Data: Relates to activity outside the organization collecting it.
• SM's sales data is internal to SM, but external to Robinson's.

Data Collection

• Census/Complete Enumeration: Method of gathering data from every unit in a population.
• Not always timely, accurate, or economical.
• Sample Survey: Gathers data from a representative population subset.
• More speedy and timely
• More economical for information gathering and analysis
• More accurate, with skilled workers making fewer errors
• Involves questions to obtain info
• Includes telephone, mailed, online, in-person and mall intercept surveys

Statistical Data Collection Methods

• Observation: Recording behavior at the time of occurrence e.g. traffic count.
• Experimental: Data collection under controlled conditions.
• Existing Studies: Using census, health statistics, etc.
  • Can be documentary sources (published reports) or field sources (researchers).
• Registration: Recording data such as car or student registration.

Sampling Techniques

• Probability Sampling: Every population element has a known chance of being selected.
• Non-Probability Sampling: Sampling procedure where not every element has a known chance of selection.
• Probability sampling is preferred for objective reliability assessment.
• Target Population: The population from which information is desired.
• Sampled Population: Collection of elements from which the sample is taken.
• Population Frame: A list of all individual units in the population.

Non-Probability Sampling Methods

• Purposive Sampling: Sample agrees with the population profile based on characteristics chosen beforehand.
• Quota Sampling: Selects specific number of units with certain characteristics.
• Convenience Sampling: Selects readily available sampling units
• Judgment Sampling: Sample selected based on expert judgment.
• Snowball/Referral Sampling: Initial subjects identify further subjects.

Probability Sampling Methods

• Simple Random Sampling: Each item has an equal chance of selection.
• Can be with replacement (SRSWR) or without replacement (SRSWOR).

Approaches to Simple Random Sampling

• Lottery Method

• Use of Random Numbers

  • Table of Random Numbers
  • Online Random Number Generator

• Stratified Random Sampling: Divides population into strata, draws a random sample from each.

• (1-in-k) Systematic Sampling: Selects every kth unit from an ordered population with a random start.

  • k is the sampling interval.

Probability Sampling Methods Continued

• Cluster Sampling: Selects distinct groups/clusters, then takes a census within selected clusters.
• Multistage Sampling: Divides the population hierarchically into sampling units across stages.

Types of Allocation

• Equal Allocation: Used when strata have similar unit numbers, variability, and sampling costs, or lack prior knowledge.
• Proportional Allocation: Used when stratum sizes vary.
• Optimum (Neyman) Allocation: Used when stratum variability or proportion varies between strata.

Data Presentation

Textual Presentation

• Data incorporation into a paragraph
• Describes figures and density

Tabular Presentation

• Systematic arrangement of data in rows/columns.
• Parts of a Formal Statistical Table:
  • Heading: Includes table number, title, and headnote.
  • Box Head: Contains column heads describing data.
  • Stub: The first column lists row captions.
  • Field: Main part of the table with the data.
  • Source Note: Cites data source.
  • Footnote: Additional notes.

Graphical Presentation

• Represents numerical values or relationships in pictorial form.

Data Description

• Aims to describe data without oversimplifying or overcomplicating.
• Data can be presented raw, as frequency distributions, or as graphs.

Measures of Central Tendency

• Central tendency identifies the "center" or typical value of a data set.
• Facilitates the comparison of two or more data sets.
• Includes the mean, median and mode

Characteristics of a Good Average

• Easily understood
• Objective and clearly defined
• Stable
• Amenable to statistical computation

Arithmetic Mean

• Sum of all values divided by the number of observations.

Population Mean Formula

• μ = (Σ Xi) / N

Sample Mean Formula

• X = (Σ Xi) / n

Characteristics of the Mean

• Uses all available information
• Influenced by extreme values and small number of observations

Mean Modifications

• May not be a value in the data set.
• Has two mathematical properties:
  • Sum of deviations from the mean is zero.
  • Sum of squared deviations is minimum.
• Always exists and is unique
• Adding/subtracting a constant changes the mean by that amount.
• Multiplying/dividing by a constant scales the mean by that constant.

Weighted Mean

• Assigns weights to observations based on their importance.
• X = (Σ WiXi) / (Σ Wi)

Combined Mean

• The combined population mean μc:
• μc = (Σ Niμi) / (Σ Ni)
• The combined sample mean Xc:
• Xc = (Σ niXi) / (Σ ni)

Trimmed Mean

• Mitigates outlier effects by deleting a% of lowest and highest values.
• The a%-trimmed mean does this then finds the mean of remaining values

Median

• Divides the data array into two equal parts.
• If n is odd, Md = X(n+1)/2
• If n is even, Md = [X(n/2) + X(n/2 + 1)] / 2

Characteristics of the Median

• Positional measure.
• Affected by item position, not value.
• Less sensitive to extreme values than the mean.

Mode

• The most frequently observed value in a data set.
• Data set can be no mode, unimodal, bimodal, trimodal, etc
• Exists if the data is high density where observation values occur
• Is not affected by extreme values
• The mode can be used for qualitative as well as quantitative data.

Choosing a Suitable measure of Central Tendency

• This depends on the data distribution and objective.

Measures of dispersion

• These describe data scatter or variability with respect to central tendency
• Necessary to describe the dataset in addition to measures of central tendency

Measures of Dispersion Definition

• Quantities indicating the extent to which individual items in a series are scattered about an average.

Measuring Dispersion

• Measures the scatter’s extent, allowing existing variation controls.
• Used to measure average value reliability

Measures of Absolute Dispersion

• Expressed in original observation units.
• Cannot compare variations when data set averages differ greatly in value/measurement units

Measures of Relative Dispersion

• Unitless, used to compare distributions.

The Range

• Difference between largest and smallest values (max - min).

Characteristics of the Range

• Uses only extreme values.
• Omits info about data clustering between extremes.
• Sensitive to outliers.

The Interquartile Range

• Difference between upper and lower quartiles (Q3 - Q1).

Characteristics of Interquartile Range

• Uses quartile values, reducing extreme value influence
• Contains middle 50% of data

The Standard Deviation and Variance

• Variance: Average squared difference from the mean.
• Standard Deviation: Square root of the variance.

Variance Formula

• Σ(Xi - μ)² / N

Standard Deviation Formula

• σ = √[Σ(Xi - μ)² / N]

Sample Variance Formula

• s² = Σ(Xi - X)² / (n-1)

Sample Standard Deviation Formula

• s = √[Σ(Xi - X)² / (n-1)]

Characteristics of Variance and Standard Deviation

• Standard deviation is used most frequently
• Variance is not a measure of absolute dispersion
• Affected by every value, skewed by extreme values
• Adding/subtracting a constant doesn't change standard deviation.
• Multip./dividing by a constant scales the standard deviation.

Coefficient of Variation

• Ratio of standard deviation to mean, in percentage.

Population Formula

• CV = (σ / μ) × 100%

Sample Formula

• CV = (s / X) × 100%

Characteristics of the Coefficient of Variation

• Compares variability even with different means/units.
• Expresses standard deviation as a percentage
• High CV means high variability
• Undefined when the mean is zero.

The Standard Score

• Z tells you how standard deviations an observation is from the mean.
• Population: Z = (X - μ) / σ,
• Sample: Z = (X - X) / s

Characteristics of the Standard Score

• Not a measure of relative dispersion itself
• useful for comparing values from different series, especially with different means, SD or units
• Helpful for outlier detection