12.2 Comparing distributions with box plots

Box plots are powerful tools for comparing distributions visually. They show key stats like median, spread, and shape at a glance. This makes it easy to spot differences between groups or datasets.

When comparing box plots, we look at center, spread, and shape. The median line shows the center, while box and whisker lengths indicate spread. Skewness and outliers reveal distribution shape. These elements help us draw meaningful conclusions about the data.

Comparing Distributions with Box Plots

Measuring Distribution Center

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• The center of a distribution is measured using the median, represented by the line inside the box of a box plot
• Comparing the position of the median lines allows for comparing the centers of multiple distributions
• Example: If the median line for Group A is higher than the median line for Group B, then the center of the distribution for Group A is higher than the center of the distribution for Group B

Assessing Distribution Spread

• The interquartile range (IQR) measures spread and represents the middle 50% of the data
  • IQR is calculated as Q3 - Q1 and is represented by the length of the box in a box plot
  • Comparing the lengths of the boxes allows for comparing the spreads of multiple distributions
• The overall range is another measure of spread that represents the distance between the minimum and maximum values, excluding outliers
  • It is represented by the distance between the whiskers in a box plot
  • Comparing the lengths of the whiskers allows for comparing the overall ranges of multiple distributions
• Example: If the box and whiskers for Group A are longer than those for Group B, then the spread of the distribution for Group A is greater than the spread of the distribution for Group B

Describing Distribution Shape

• The shape of a distribution can be described as symmetric, left-skewed, or right-skewed
  • In a box plot, a symmetric distribution has the median line in the center of the box and whiskers of equal length
  • A left-skewed distribution has a longer lower whisker and more data on the left side of the median
  • A right-skewed distribution has a longer upper whisker and more data on the right side of the median
• Outliers are data points that are far from the rest of the distribution and are represented by individual points beyond the whiskers in a box plot
  • The presence and position of outliers can impact the interpretation of the distribution's shape and spread
• Example: If a box plot has a longer upper whisker and several outliers on the right side, the distribution is likely right-skewed

Statistical Significance of Differences

Assessing Statistical Significance

• Statistical significance refers to the likelihood that observed differences between distributions are due to chance rather than a real difference in the populations
  • It is typically assessed using a p-value, which represents the probability of observing the data if the null hypothesis (no real difference) is true
• The significance level (α) is the threshold for determining statistical significance
  • It represents the maximum acceptable probability of rejecting the null hypothesis when it is actually true (Type I error)
  • Common significance levels are 0.01, 0.05, and 0.10
• If the p-value is less than the significance level, the difference is considered statistically significant, and the null hypothesis is rejected in favor of the alternative hypothesis
• If the p-value is greater than the significance level, the difference is not considered significant, and the null hypothesis is not rejected

Conducting Hypothesis Tests

• Hypothesis testing is a statistical method used to determine if differences between distributions are significant
  • It involves stating a null hypothesis (H0) and an alternative hypothesis (Ha), setting a significance level (α), and calculating a test statistic and p-value based on the data
• The choice of hypothesis test depends on the type of data and the assumptions made about the populations
  • Common tests for comparing distributions include the two-sample t-test (for comparing means of normally distributed data), the Wilcoxon rank-sum test (for comparing medians of non-normally distributed data), and the chi-square test (for comparing proportions of categorical data)
• Example: To compare the mean heights of two groups, a researcher might use a two-sample t-test with a significance level of 0.05. If the resulting p-value is 0.02, the difference in mean heights would be considered statistically significant

Sample Size Impact on Box Plots

Effect of Sample Size on Variability

• Sample size refers to the number of observations in a dataset
  • Larger sample sizes generally provide more precise estimates of population parameters and are less affected by extreme values or outliers
• As sample size increases, the variability of the sample statistics (such as the median and IQR) decreases, leading to narrower boxes and whiskers in the box plot
  • This is because larger samples are more likely to be representative of the population, and extreme values have less impact on the overall distribution
• Small sample sizes can lead to more variability in the sample statistics and wider boxes and whiskers in the box plot
  • This is because small samples are more likely to be influenced by extreme values or outliers, which can distort the appearance of the distribution

Considerations for Comparing Box Plots

• When comparing box plots with different sample sizes, it is important to consider the potential impact of sample size on the observed differences
  • Differences that appear large in small samples may not be statistically significant, while small differences in large samples may be significant
• The choice of sample size depends on factors such as the variability of the population, the desired level of precision, and the available resources
  • Increasing the sample size can improve the precision and reliability of the results but may also increase the cost and time required for data collection and analysis
• Example: If a researcher compares the box plots of test scores for a class of 20 students and a class of 200 students, the box plot for the larger class will likely have narrower boxes and whiskers due to the increased sample size

Population Conclusions from Samples

Inferring Population Differences

• Inferential statistics involves using sample data to make conclusions about the larger population from which the samples were drawn
  • When comparing distributions of sample data using box plots, the goal is often to infer differences or similarities between the corresponding populations
• The central limit theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution
  • This allows for using statistical methods based on normal distributions, such as the t-test, to compare means of samples and make inferences about the populations
• Confidence intervals provide a range of plausible values for a population parameter based on the sample data
  • They can be used to estimate the difference between population parameters (such as medians) based on the differences observed in the samples
  • If the confidence interval for the difference includes zero, the difference is not considered statistically significant

Limitations and Biases in Sampling

• When drawing conclusions about populations based on sample comparisons, it is important to consider the limitations and potential biases of the sampling method
  • Random sampling, where each member of the population has an equal chance of being selected, is ideal for making unbiased inferences
  • Non-random sampling methods, such as convenience or voluntary response sampling, can introduce bias and limit the generalizability of the results
• The scope of inference refers to the population or setting to which the conclusions can be applied
  • When comparing distributions from different populations or settings, it is important to consider the similarity of the populations and the potential for confounding variables that could explain the observed differences
  • Conclusions should be limited to the specific populations and settings represented by the samples
• Example: If a researcher compares the box plots of income for a random sample of households in two cities and finds a significant difference, they might conclude that the median income differs between the two populations. However, if the samples were not truly random or representative, the conclusion may not be valid for the entire populations of the cities