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### 6.9: Uniform Distribution TABLE OFCONTENTS X ## Chapter 1: Understanding Statistics 301.1: Introduction to Statistics301.2: How Data are Classified: Categorical Data301.3: How Data are Classified: Numerical Data301.4: Nominal Level of Measurement301.5: Ordinal Level of Measurement301.6: Interval Level of Measurement301.7: Ratio Level of Measurement301.8: Data Collection by Observations301.9: Data Collection by Experiments301.10: Data Collection by Survey301.11: Random Sampling Method301.12: Systematic Sampling Method301.13: Convenience Sampling Method301.14: Stratified Sampling Method301.15: Cluster Sampling Method ## Chapter 2: Summarizing and Visualizing Data 302.1: Review and Preview302.2: What is a Frequency Distribution302.3: Construction of Frequency Distribution302.4: Relative Frequency Distribution302.5: Percentage Frequency Distribution302.6: Cumulative Frequency Distribution302.7: Ogive Graph302.8: Histogram302.9: Relative Frequency Histogram302.10: Scatter Plot302.11: Time-Series Graph302.12: Bar Graph302.13: Multiple Bar Graph302.14: Pareto Chart302.15: Pie Chart ## Chapter 3: Measure of Central Tendency 303.1: What is Central Tendency?303.2: Arithmetic Mean303.3: Geometric Mean303.4: Harmonic Mean303.5: Trimmed Mean303.6: Weighted Mean303.7: Root Mean Square303.8: Mean From a Frequency Distribution303.9: What is a Mode?303.10: Median303.11: Midrange303.12: Skewness303.13: Types of Skewness ## Chapter 4: Measures of Variation 304.1: What is Variation?304.2: Range304.3: Standard Deviation304.4: Standard Error of the Mean304.5: Calculating Standard Deviation304.6: Variance304.7: Coefficient of Variation304.8: Range Rule of Thumb to Interpret Standard Deviation304.9: Empirical Method to Interpret Standard Deviation304.10: Chebyshev's Theorem to Interpret Standard Deviation304.11: Mean Absolute Deviation ## Chapter 5: Measures of Relative Standing 305.1: Review and Preview305.2: Introduction to z Scores305.3: z Scores and Unusual Values305.4: Percentile305.5: Quartile305.6: 5-Number Summary305.7: Boxplot305.8: What Are Outliers?305.9: Modified Boxplots ## Chapter 6: Probability Distributions 306.1: Probability in Statistics306.2: Random Variables306.3: Probability Distributions306.4: Probability Histograms306.5: Unusual Results306.6: Expected Value306.7: Binomial Probability Distribution306.8: Poisson Probability Distribution306.9: Uniform Distribution306.10: Normal Distribution306.11: z Scores and Area Under the Curve306.12: Applications of Normal Distribution306.13: Sampling Distribution306.14: Central Limit Theorem ## Chapter 7: Estimates 307.1: What are Estimates?307.2: Sample Proportion and Population Proportion307.3: Confidence Intervals307.4: Confidence Coefficient307.5: Interpretation of Confidence Intervals307.6: Critical Values307.7: Margin of Error307.8: Sample Size Calculation307.9: Estimating Population Mean with Known Standard Deviation307.10: Estimating Population Mean with Unknown Standard Deviation307.11: Confidence Interval for Estimating Population Mean ## Chapter 8: Distributions 308.1: Distributions to Estimate Population Parameter308.2: Degrees of Freedom308.3: Student t Distribution308.4: Choosing Between z and t Distribution308.5: Chi-square Distribution308.6: Finding Critical Values for Chi-Square308.7: Estimating Population Standard Deviation308.8: Goodness-of-Fit Test308.9: Expected Frequencies in Goodness-of-Fit Tests308.10: Contingency Table308.11: Introduction to Test of Independence308.12: Hypothesis Test for Test of Independence308.13: Determination of Expected Frequency308.14: Test for Homogeneity308.15: F Distribution ## Chapter 9: Hypothesis Testing 309.1: What is a Hypothesis?309.2: Null and Alternative Hypotheses309.3: Critical Region, Critical Values and Significance Level309.4: P-value309.5: Types of Hypothesis Testing309.6: Decision Making: P-value Method309.7: Decision Making: Traditional Method309.8: Hypothesis: Accept or Fail to Reject?309.9: Errors In Hypothesis Tests309.10: Testing a Claim about Population Proportion309.11: Testing a Claim about Mean: Known Population SD309.12: Testing a Claim about Mean: Unknown Population SD309.13: Testing a Claim about Standard Deviation ## Chapter 10: Analysis of Variance 3010.1: What is an ANOVA?3010.2: One-Way ANOVA3010.3: One-Way ANOVA: Equal Sample Sizes3010.4: One-Way ANOVA: Unequal Sample Sizes3010.5: Multiple Comparison Tests3010.6: Bonferroni Test3010.7: Two-Way ANOVA ## Chapter 11: Correlation and Regression 3011.1: Correlation3011.2: Coefficient of Correlation3011.3: Calculating and Interpreting the Linear Correlation Coefficient3011.4: Regression Analysis3011.5: Outliers and Influential Points3011.6: Residuals and Least-Squares Property3011.7: Residual Plots3011.8: Variation3011.9: Prediction Intervals3011.10: Multiple Regression ## Chapter 12: Statistics in Practice 3012.1: What is an Experiment?3012.2: Study Design in Statistics3012.3: Observational Studies3012.4: Experimental Designs3012.5: Randomized Experiments3012.6: Crossover Experiments3012.7: Controls in Experiments3012.8: Bias3012.9: Blinding3012.10: Clinical Trials Full Table of Contents

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Uniform Distribution

### 6.9: Uniform Distribution

The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.

Two essential properties of this distribution are

1. The area under the rectangular shape equals 1.
2. There is a correspondence between the probability of an event and the area under the curve.

Further, the mean and standard deviation of the uniform distribution can be calculated when the lower and upper cut-offs, denoted as a and b, respectively, are given. For a random variable x, in a uniform distribution, given a and b, the probability density function is f(x) is calculated as Consider data of 55 smiling times, in seconds, of an eight-week-old baby:

10.4, 19.6, 18.8, 13.9, 17.8, 16.8, 21.6, 17.9, 12.5, 11.1, 4.9, 12.8, 14.8, 22.8, 20.0, 15.9, 16.3, 13.4, 17.1, 14.5, 19.0, 22.8, 1.3, 0.7, 8.9, 11.9, 10.9, 7.3, 5.9, 3.7, 17.9, 19.2, 9.8, 5.8, 6.9, 2.6, 5.8, 21.7, 11.8, 3.4, 2.1, 4.5, 6.3, 10.7, 8.9, 9.4, 9.4, 7.6, 10.0, 3.3, 6.7, 7.8, 11.6, 13.8 and, 18.6. Assume that the smiling times follow a uniform distribution between zero and 23 seconds, inclusive. Note that zero and 23 are the lower and upper cut-offs for the uniform distribution of smiling times.

Since the smiling times' distribution is a uniform distribution, it can be said that any smiling time from zero to and including 23 seconds has an equal likelihood of occurrence. A histogram that can be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.

For this example, the random variable, x  = length, in seconds, of an eight-week-old baby's smile. The notation for the uniform distribution is x ~ U(a, b) where a = the lowest value (lower cut-off) of x and b = the highest value (upper cut-off) of x. For this example, a = 0 and b = 23.

The mean, μ, is calculated using the following equation: The mean for this distribution is 11.50 seconds. The smile of an eight-week-old baby lasts for an average time of 11.50 seconds.

The standard deviation, σ, is calculated using the formula: The standard deviation for this example is 6.64 seconds.

This text is adapted from Openstax, Introductory Statistics, Section 5.2 The Uniform Distribution

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