Percentiles and Z scores

Analysis of Data in Kinesiology I

  • Topic 5: Percentiles and Z-Scores

  • Course: KINE 2050

  • Term: Winter 2024

Standard Normal Distribution

  • Characteristics of Distribution:

    • Mean (μ) = 0

    • Standard Deviation (σ) = 1

  • Key Percentages:

    • 68.3% of data lies within ±1σ

    • 95.5% within ±2σ

    • 99.7% within ±3σ

  • Visual Representation: Depicts a graphical distribution showing the percentage of values falling within certain standard deviations from the mean.

Inter-Quartile Range (IQR)

  • Definition: Measure of variability that divides a dataset into quartiles.

  • Quartiles Breakdown:

    • First Quartile (Q1): 25th percentile

    • Second Quartile (Q2): 50th percentile (Median)

    • Third Quartile (Q3): 75th percentile

  • Calculation Example:

    • For dataset: 62, 63, 64, 64, 70, 72, 76, 77, 81, 81

    • IQR = Q3 - Q1 = 77 - 64 = 13

    • Q1 = 64, Q3 = 77, Median = 72.

Percentiles

  • Definition: The pth percentile indicates the value below which p% of the data fall.

  • Quartiles:

    • Q1: 25th percentile

    • Q2: 50th percentile (Median)

    • Q3: 75th percentile

  • Example Calculation: Given a dataset, determine how many scores fall below a certain raw score to find the corresponding percentile.

Relative Scores Using Percentiles and Z-Scores

  • Represents an individual's standing relative to a group.

  • Transformations: Raw scores can be converted to either percentiles or z-scores for comparison.

  • Application: Useful in various domains such as fitness assessments, exams, and health data.

Calculating Percentiles

  • Process: To find a percentile score for a raw data point,

    1. Order the scores.

    2. Locate the score.

    3. Calculate percentile as:P = (rank of score / total scores) × 100.

  • Example: For scores of 2, 9, 6, 5, 16, 15, 10, 8, 7, 4, 1, calculate rank and resulting percentile.

Grouped Frequency Distribution

  • Understanding Percentiles: Calculation for grouped data involves finding the raw score corresponding to a certain percentile.

  • Formula for Percentile:

    • Given class intervals with their corresponding frequencies, apply:P = LL + (P * N - Σf) / f * iwhere LL is the lower limit, P is the desired percentile, and N is total frequency.

  • Example Calculation: To find percentile for score 0.32 in grouped data with an interval size plotted against class frequency.

Z-Scores

  • Definition: A z-score indicates how many standard deviations a data point is from the mean.

  • Formula:Z = (x - μ) / σwhere x = raw score, μ = mean, σ = standard deviation

  • Uses of Z-Scores:

    1. Compare individual scores to a group.

    2. Standardize scores across different measures.

    3. Calculate probabilities based on standard normal curve.

  • **Example Calculations:

    1. Given a score of 6 with a mean of 5 and standard deviation of 1: Z = (6 - 5)/1 = +1.

    2. Conversion of raw scores into Z-scores to visualize standings.

Applications of Z-Scores

  • Allows understanding of position within a normal distribution.

  • Useful in determining probabilities and interpreting statistics in various disciplines.

  • Example: Calculate the percentage of a group that exceeds a certain raw score using the z-score's cumulative area under the curve.

Class Practice Questions

  1. Determine the percentile for a score in given datasets.

  2. Calculate z-scores for specified conditions and find corresponding percentiles.

  3. Evaluate the frequency of individuals falling within a Z-score range.

Summary of Z-Scores and Percentiles

  • Z-scores and percentiles act as standardized measures to compare relative standings.

  • Essential tools in statistics, health data evaluations, and academic assessments for guiding interpretations according to group distributions.