(STATS) 2025 MMW
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Outline of Topics: Measures of Central Tendency
Lesson: Measures of Dispersion Overview
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Quote: "Statistics is the grammar of science." - Karl Pearson
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Definition: Statistics comes from Latin "status" or Italian "statista," meaning "political state" or "government."
It deals with collection, presentation, analysis, and interpretation of data.
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Types of Statistics
Descriptive Statistics: Gathering, classification, and presentation of data to summarize group characteristics.
Inferential Statistics: Making inferences or predictions about a large set of data using gathered information.
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Data: Individual pieces of factual information recorded for analysis; refers to organized sets of values by variables.
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Types of Data
Quantitative Data: Measurable with numbers (e.g., speed, duration).
Discrete: Whole numbers (e.g., count).
Continuous: Can be broken down (e.g., height, weight).
Qualitative Data: Non-numerical, categorical data (e.g., yes/no, eye color).
Nominal: for naming variables.
Ordinal: describes order (e.g., rankings).
Interval: known differences (e.g., temperature).
Ratio: measurable intervals (e.g., weight).
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The Four Scales of Measurement
Nominal Scale: Used for naming variables without order.
Ordinal Scale: Ranked order without determined differences.
Interval Scale: Numerical variables with equal intervals.
Ratio Scale: Variables with measurable intervals.
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Sample Size & Representation
Population (N): Total items in a group
Sample (n): Subset of the population
Slovin's Formula: n = N / (1 + Ne²)
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Sampling Techniques
Simple Random Sample
Systematic Sample
Stratified Sample
Cluster Sample
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Measures of Central Tendency
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Measures of Central Tendency
Mean (x̄)
Median
Mode
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Mean (Ungrouped Data)
Formula: x̄ = Σx / n
Represents the center of gravity in a distribution.
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Example: Calculating Mean
Given Data: 65, 55, 89, 56, 35, 14, 56, 55, 87, 45, 92
Mean Calculation: Sum = 645; n = 11; Mean = 59
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Mean (Grouped Data)
Class intervals with frequency are used to find mean: x̄ = Σfx / n
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Median (Ungrouped Data)
Definition: Positional value or midpoint.
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Median (Grouped Data)
Formula: Median = LLR + (n-F)/f * i
LLR = lower limit, F = cumulative frequency, n = sample size, f = frequency.
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Example: Finding Median
Given Data: 65, 55, 89, 56, 35, 14, 56, 55, 87, 45, 92
Arranged Data: 14, 35, 45, 55, 55, 56, 56, 65, 87, 89, 92
Median = 56
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Example: Median (Grouped Data)
Class intervals and frequencies calculated to find median value.
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Mode (Ungrouped Data)
Definition: Most frequent value in a dataset.
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Mode (Grouped Data)
Formula: Mode = LLR + (du/(du+dl)) * i
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Example: Finding Mode
Given Data: 65, 55, 89, 56, 35, 14, 56, 55, 55, 87, 92
Mode = 55
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Example: Mode (Grouped Data)
Class intervals and frequency used to compute mode.
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Measures of Dispersion
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Range
Simplest measure of dispersion, calculated as: Range = Highest score - Lowest score.
Example: Range = 92 - 14 = 78
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Variance
Variance (ungrouped data): Measure of variability considering the position of observations relative to the mean:
Formula: Variance = S² = Σ(x - x̄)² / (n - 1)
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Example: Variance Calculation
Provided calculations show variance for ungrouped data.
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Standard Deviation
Defined as the positive square root of variance.
Represents the standard unit for measuring distances of scores from the mean.
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Example: Standard Deviation
Detailed calculations presented for finding standard deviation of a sample.
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Formulas
Summation of various formulas related to standard deviation and variance for different categories.
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Thank You!