Name: Introduction to z Scores
Uploaded: 2022-12-05T09:15:22-05:00
Duration: 1 min 6 s
Description: A z score (or standardized value) is measured in units of the standard deviation. It tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ.

Chapter 5: Measures of Relative Standing Back To Chapter

5.2: Introduction to z Scores

TABLE OF
CONTENTS

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Chapter 1: Understanding Statistics

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1.1: Introduction to Statistics
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1.2: How Data are Classified: Categorical Data
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1.3: How Data are Classified: Numerical Data
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1.4: Nominal Level of Measurement
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1.5: Ordinal Level of Measurement
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1.6: Interval Level of Measurement
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1.7: Ratio Level of Measurement
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1.8: Data Collection by Observations
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1.9: Data Collection by Experiments
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1.10: Data Collection by Survey
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1.11: Random Sampling Method
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1.12: Systematic Sampling Method
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1.13: Convenience Sampling Method
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1.14: Stratified Sampling Method
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1.15: Cluster Sampling Method

Chapter 2: Summarizing and Visualizing Data

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2.1: Review and Preview
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2.2: What is a Frequency Distribution
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2.3: Construction of Frequency Distribution
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2.4: Relative Frequency Distribution
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2.5: Percentage Frequency Distribution
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2.6: Cumulative Frequency Distribution
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2.7: Ogive Graph
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2.8: Histogram
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2.9: Relative Frequency Histogram
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2.10: Scatter Plot
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2.11: Time-Series Graph
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2.12: Bar Graph
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2.13: Multiple Bar Graph
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2.14: Pareto Chart
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2.15: Pie Chart

Chapter 3: Measure of Central Tendency

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3.1: What is Central Tendency?
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3.2: Arithmetic Mean
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3.3: Geometric Mean
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3.4: Harmonic Mean
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3.5: Trimmed Mean
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3.6: Weighted Mean
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3.7: Root Mean Square
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3.8: Mean From a Frequency Distribution
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3.9: What is a Mode?
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3.10: Median
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3.11: Midrange
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3.12: Skewness
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3.13: Types of Skewness

Chapter 4: Measures of Variation

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4.1: What is Variation?
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4.2: Range
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4.3: Standard Deviation
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4.4: Standard Error of the Mean
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4.5: Calculating Standard Deviation
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4.6: Variance
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4.7: Coefficient of Variation
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4.8: Range Rule of Thumb to Interpret Standard Deviation
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4.9: Empirical Method to Interpret Standard Deviation
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4.10: Chebyshev's Theorem to Interpret Standard Deviation
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4.11: Mean Absolute Deviation

Chapter 5: Measures of Relative Standing

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5.1: Review and Preview
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5.2: Introduction to z Scores
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5.3: z Scores and Unusual Values
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5.4: Percentile
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5.5: Quartile
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5.6: 5-Number Summary
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5.7: Boxplot
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5.8: What Are Outliers?
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5.9: Modified Boxplots

Chapter 6: Probability Distributions

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6.1: Probability in Statistics
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6.2: Random Variables
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6.3: Probability Distributions
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6.4: Probability Histograms
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6.5: Unusual Results
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6.6: Expected Value
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6.7: Binomial Probability Distribution
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6.8: Poisson Probability Distribution
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6.9: Uniform Distribution
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6.10: Normal Distribution
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6.11: z Scores and Area Under the Curve
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6.12: Applications of Normal Distribution
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6.13: Sampling Distribution
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6.14: Central Limit Theorem

Chapter 7: Estimates

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7.1: What are Estimates?
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7.2: Sample Proportion and Population Proportion
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7.3: Confidence Intervals
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7.4: Confidence Coefficient
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7.5: Interpretation of Confidence Intervals
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7.6: Critical Values
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7.7: Margin of Error
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7.8: Sample Size Calculation
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7.9: Estimating Population Mean with Known Standard Deviation
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7.10: Estimating Population Mean with Unknown Standard Deviation
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7.11: Confidence Interval for Estimating Population Mean

Chapter 8: Distributions

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8.1: Distributions to Estimate Population Parameter
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8.2: Degrees of Freedom
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8.3: Student t Distribution
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8.4: Choosing Between z and t Distribution
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8.5: Chi-square Distribution
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8.6: Finding Critical Values for Chi-Square
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8.7: Estimating Population Standard Deviation
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8.8: Goodness-of-Fit Test
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8.9: Expected Frequencies in Goodness-of-Fit Tests
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8.10: Contingency Table
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8.11: Introduction to Test of Independence
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8.12: Hypothesis Test for Test of Independence
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8.13: Determination of Expected Frequency
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8.14: Test for Homogeneity
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8.15: F Distribution

Chapter 9: Hypothesis Testing

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9.1: What is a Hypothesis?
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9.2: Null and Alternative Hypotheses
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9.3: Critical Region, Critical Values and Significance Level
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9.4: P-value
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9.5: Types of Hypothesis Testing
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9.6: Decision Making: P-value Method
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9.7: Decision Making: Traditional Method
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9.8: Hypothesis: Accept or Fail to Reject?
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9.9: Errors In Hypothesis Tests
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9.10: Testing a Claim about Population Proportion
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9.11: Testing a Claim about Mean: Known Population SD
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9.12: Testing a Claim about Mean: Unknown Population SD
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9.13: Testing a Claim about Standard Deviation

Chapter 10: Analysis of Variance

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10.1: What is an ANOVA?
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10.2: One-Way ANOVA
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10.3: One-Way ANOVA: Equal Sample Sizes
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10.4: One-Way ANOVA: Unequal Sample Sizes
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10.5: Multiple Comparison Tests
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10.6: Bonferroni Test
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10.7: Two-Way ANOVA

Chapter 11: Correlation and Regression

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11.1: Correlation
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11.2: Coefficient of Correlation
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11.3: Calculating and Interpreting the Linear Correlation Coefficient
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11.4: Regression Analysis
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11.5: Outliers and Influential Points
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11.6: Residuals and Least-Squares Property
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11.7: Residual Plots
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11.8: Variation
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11.9: Prediction Intervals
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11.10: Multiple Regression

Chapter 12: Statistics in Practice

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12.1: What is an Experiment?
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12.2: Study Design in Statistics
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12.3: Observational Studies
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12.4: Experimental Designs
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12.5: Randomized Experiments
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12.6: Crossover Experiments
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12.7: Controls in Experiments
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12.8: Bias
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12.9: Blinding
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12.10: Clinical Trials

Full Table of Contents

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Introduction to z Scores

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TRANSCRIPT

The z score, or standardized score, is the number of standard deviations that a given value is away from the mean. It is one of the commonly used measures of relative standing.

Using the standardization formula, data can be converted into corresponding z scores. The standardization formula for a population and a sample differ in the mean and standard deviation notations.

z scores provide the relative position of a data point. A positive z score means the data point is above the mean and a negative z indicates below the mean. The mean value always has a zero z score.

The z score is also used to compare data measured on different scales, such as comparing a student’s height and weight with classmates. Since data are measured on different scales, they are standardized into z scores.

A z score of 1.5 indicates that the student is taller than most of his classmates, while a z score of minus 0.5 suggests that his weight is very close to the class average.

5.2: Introduction to z Scores

A z score (or standardized value) is measured in units of the standard deviation. It tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a zero z score. It is important to note that the mean of the z scores is zero, and the standard deviation is one.

z scores help to find the outliers or unusual values from any data distribution. According to the range rule of thumb, outliers or unusual values have z scores less than -2 or greater than +2.

This text is adapted from Openstax, Introductory Statistics, 6.1 The Standard Normal Distribution

Chapter 1: Understanding Statistics

Chapter 2: Summarizing and Visualizing Data

Chapter 3: Measure of Central Tendency

Chapter 4: Measures of Variation

Chapter 5: Measures of Relative Standing

Chapter 6: Probability Distributions

Chapter 7: Estimates

Chapter 8: Distributions

Chapter 9: Hypothesis Testing

Chapter 10: Analysis of Variance

Chapter 11: Correlation and Regression

Chapter 12: Statistics in Practice

5.2: Introduction to z Scores

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