Elementary Stats Lesson #4

The instructor begins by introducing the concept of descriptive statistics, specifically focusing on measures of position within a data set. The lesson aims to explain how to use numerical values to describe the position of an observation relative to the rest of the data. The concept of standardizing data points through z-scores is introduced, which involves calculating how many standard deviations an observation is from the mean. The z-score formula is provided for both sample data and population data. The empirical rule is also discussed, which is applicable to symmetric, bell-shaped distributions, and it describes the distribution of data within standard deviations from the mean.

This paragraph delves deeper into the use of z-scores, explaining their importance in statistical analysis and inference. Z-scores are particularly useful for comparing data points from different data sets, effectively allowing for the comparison of 'apples and oranges' by standardizing measurements. The instructor provides an example comparing the heights of adult men and women in the U.S., using z-scores to determine who is relatively taller within their respective populations. The paragraph emphasizes the growing relevance of z-scores as the course progresses.

The instructor introduces percentiles and quartiles as measures of position within a data set. Percentiles divide the data in a way that a specific percentage of observations fall below a certain value. Quartiles are special cases of percentiles, representing the 25th, 50th, and 75th percentiles. An algorithm for finding any percentile is presented, involving sorting the data, calculating the location of the percentile, and determining the actual value based on whether the calculated location is a whole number or not. The paragraph also explains how to find quartiles for a given data set of SAT math scores.

The concept of outliers in a data set is discussed, along with the resistance of quartiles to the impact of outliers and skewness. The interquartile range (IQR) is introduced as a measure of spread for the middle 50% of the data. The instructor explains how to calculate the IQR and provides a method for identifying outliers using the IQR, known as the IQR test. This involves calculating the lower and upper fences and comparing them to the data points to determine if any are outliers.

The five number summary is introduced as a package for summarizing data, which includes the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. This method is particularly useful for data sets with outliers or skewed distributions. The paragraph explains how to use a calculator to obtain these values and describes the box plot as a graphical representation of the five number summary. The instructor also discusses the modified box plot, which is helpful for identifying outliers without performing the IQR test.

The instructor discusses the concept of resistance in numerical summaries, highlighting that the median and mode are resistant to the influence of outliers and skewness, while the mean is not. The paragraph emphasizes the importance of choosing the appropriate measures of center and spread based on the characteristics of the data set, such as symmetry and the presence of outliers. The median and interquartile range (IQR) are recommended for skewed distributions or those with outliers, whereas the mean and standard deviation are suitable for symmetric, bell-shaped distributions.

The instructor demonstrates how to create a box plot using a calculator, using the SAT math scores as an example. The process involves storing the data, selecting the modified box plot option, and allowing the calculator to draw the plot. The box plot visually represents the five number summary and can indicate outliers. The paragraph also describes what to look for in a box plot to determine if a data set is symmetric, right-skewed, or left-skewed, based on the position of the median within the IQR box and the comparative lengths of the whiskers.

In the final paragraph, the instructor wraps up the chapter on descriptive statistics by summarizing the key points discussed, including the importance of understanding resistance in numerical summaries. The paragraph emphasizes the differences between the mean, median, and mode in the context of symmetric and skewed distributions. The instructor also highlights the ability to perform various statistical analyses using technology, such as the TI-83 or 84 calculator, which can automate the process of obtaining summaries and visualizing data with box plots and histograms.