2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends, that is, finding a general pattern in data sets, including temperature, sales, employment, company profit, or cost, over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Bar graphs are especially useful when categorical data are being used.

2.2 Histograms, Frequency Polygons, and Time Series Graphs

A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values, and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually go on the y-axis with the frequency being graphed on the x-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

2.3 Measures of the Location of the Data

The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50^th percentile would be greater than 50 percent of the other observations in the set. Quartiles divide data into quarters. The first quartile (Q₁) is the 25^th percentile, the second quartile (Q₂ or median) is the 50^th percentile, and the third quartile (Q₃) is the 75^th percentile. The interquartile range, or IQR, is the range of the middle 50 percent of the data values. The IQR is found by subtracting Q₁ from Q₃ and can help determine outliers by using the following two expressions.

• Q₃ + IQR(1.5)
• Q₁ – IQR(1.5)

2.4 Box Plots

Box plots are a type of graph that can help visually organize data. Before a box plot can be graphed, the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data.

2.5 Measures of the Center of the Data

The mean and the median can be calculated to help you find the center of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum (or data) in your data set. The mean, median, and mode are extremely helpful when you need to analyze your data, but if your data set consists of ranges that lack specific values, the mean may seem impossible to calculate. However, the mean can be approximated if you add the lower boundary with the upper boundary and divide by two to find the midpoint of each interval. Multiply each midpoint by the number of values found in the corresponding range. Divide the sum of these values by the total number of data values in the set.

2.6 Skewness and the Mean, Median, and Mode

Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like Figure 2.19. A left (or negative) skewed distribution has a shape like Figure 2.20. A symmetrical distribution looks like Figure 2.18.

2.7 Measures of the Spread of the Data

The standard deviation can help you calculate the spread of data. There are different equations to use if you are calculating the standard deviation of a sample or of a population.

• The standard deviation allows us to compare individual data or classes to the data set mean numerically.
• s = $\sqrt{\frac{{{\displaystyle \sum }}^{\text{}}{(x-\overline{x})}^2}{n-1}}$ or s = $\sqrt{\frac{{{\displaystyle \sum }}^{\text{}}f{(x-\overline{x})}^2}{n-1}}$ is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ, and the formula σ = $\sqrt{\frac{{{\displaystyle \sum }}^{\text{}}{(x-\mu )}^2}{N}}$ or σ = $\sqrt{\frac{{{\displaystyle \sum }}^{\text{}}f{(x-\mu )}^2}{N}}$.