Diagrams and Graphs of Statistical Data

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Steven Daniel Roberts
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1 Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in which statistical data may be displayed pictorially such as different types of graphs and diagrams. 1

2 Patterns in Data Graphic displays are useful for seeing patterns in data. Patterns in data are commonly described in terms of: center, spread, shape, and unusual features. Some common distributions have special descriptive labels, such as: symmetric, bell-shaped, skewed, etc. 2

3 Center Graphically, the center of a distribution is located at the median of the distribution. This is the point in a graphic display where about half of the observations are on either side. In the chart to the right, the height of each column indicates the frequency of observations. Here, the observations are centered over 4. 3

This is the point in a graphic display where about half of the observations are on

4 Spread The spread of a distribution refers to the variability of the data. If the observations cover a wide range, the spread is larger. If the observations are clustered around a single value, the spread is smaller. Consider the figures above. In the figure on the left, data values range from 3 to 7; whereas in the figure on the right, values range from 1 to 9. The figure on the right is more variable, so it has the greater spread. 4

If the observations are clustered around a single value, the spread is smaller. Consider the figures above.

5 Shape The shape of a distribution is described by the following characteristics. 1. Symmetry. When it is graphed, a symmetric distribution can be divided at the center so that each half is a mirror image of the other. 2. Number of peaks. Distributions can have few or many peaks. Distributions with one clear peak are called unimodal, and distributions with two clear peaks are called bimodal. When a symmetric distribution has a single peak at the center, it is referred to as bell-shaped. 3. Skewness. When they are displayed graphically, some distributions have many more observations on one side of the graph than the other. Distributions with most of their observations on the left (toward lower values) are said to be skewed right; and distributions with most of their observations on the right (toward higher values) are said to be skewed left. 4. Uniform. When the observations in a set of data are equally spread across the range of the distribution, the distribution is called a uniform distribution. A uniform distribution has no clear peaks. 5

Distributions with one clear peak are called unimodal, and distributions with two clear peaks are called bimodal.

6 Some examples of distributions and shapes. 6

7 Unusual Features Sometimes, statisticians refer to unusual features in a set of data. The two most common unusual features are gaps and outliers. Gaps. Gaps refer to areas of a distribution where there are no observations. The first figure below has a gap; there are no observations in the middle of the distribution. Outliers. Sometimes, distributions are characterized by extreme values that differ greatly from the other observations. These extreme values are called outliers. The second figure below illustrates a distribution with an outlier. Except for one lonely observation (the outlier on the extreme right), all of the observations fall between 0 and 4. 7

Outliers. Sometimes, distributions are characterized by extreme values that differ greatly from the other observations. These extreme values are called outliers.

8 Graphs in Statistics 1. Bar graph 2. Histogram 3. Pie graph 4. Line graph 5. Boxplot graph 6. Scatter graph 8

9 Bar Charts A bar graph is a way to visually represent qualitative data. A bar chart is made up of columns plotted on a graph. Here is how to read a bar chart. The columns are positioned over a label that represents a categorical variable. The height of the column indicates the size of the group defined by the column label. 9

10 Frecvency table

11 Histograms Histograms are graphs of a distribution of data designed to show centering, dispersion (spread), and shape (relative frequency) of the data. Like a bar chart, a histogram is made up of columns plotted on a graph. Usually, there is no space between adjacent columns. The columns are positioned over a label that represents a quantitative variable. The column label can be a single value or a range of values. The height of the column indicates the size of the group defined by the column label. 11

Usually, there is no space between adjacent columns.

12 Frecvency table

14 Problem Consider the histograms below. Which of the following statements are true? I. Both data sets are symmetric. II. Both data sets have the same range. (A) I only (B) II only (C) I and II (D) Neither is true. (E) There is insufficient information to answer this question.

Both data sets are symmetric. II. Both data sets have the same range.

15 Pie Chart Pie Chart or Circle Graph - A pie chart displays qualitative data in the form of a pie. Each slice of pie represents a different category. In a pie chart, the arc length of each sector (and consequently its central angle and area), is proportional to the quantity it represents. 15

16 Pie chart Eye colors of 100 third grader students. Brown corresponds to brown eyes, blue to blue eyes, and green to hazel eyes. A pie chart is a way of summarizing a set of categorical data. It is a circle which is divided into segments. Each segment represents a particular category. The area of each segment is proportional to the number of cases in that category. 16

A pie chart is a way of summarizing a set of categorical data.

17 Haw to create a pie chart. Expenditure Items Expenditure Angle of sectors Cumulative angle Food Clothing House rent Fuel and Lighting Miscellaneous Total Food House rent Miscellaneous Clothing Fuel and Lighting 17

Clothing 100 24 168 House rent 400 96 264 Fuel and Lighting 100 24 288

18 Line graph 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. 18

That is, finding a general pattern in data sets including temperature, sales,

19 Line graph. Exemple A cylinder of liquid was heated. Its temperature was recorded at ten-minute intervals as shown in the following table Time in minutes Temperature in C a. Draw a line graph to represent this information. b. Estimate the temperature of the cylinder after 25 minutes of heating. 19

Time in minutes Temperature in C 0 5 10 26 20 45 30 61 40 74 50 80 60 85 a.

20 Boxplot graph What is a box plot? A box plot is a diagram that gives a visual representation to the distribution of the data, highlighting where most values lie and those values that greatly differ from the norm, called outliers. The box plot is also referred to as box and whisker plot or box and whisker diagram 20

of the data, highlighting where most values lie and those values that greatly

Elements of the box plot http://www.youtube.com/watch?

21 Elements of the box plot

22 Consider the boxplot below. Which of the following statements are true? I. The distribution is skewed right. II. The interquartile range is about 8. III. The median is about 10. (A) I only (B) II only (C) III only (D) I and III (E) II and III 22

23 Scatter graph A scatterplot is a graphic tool used to display the relationship between two quantitative variables. It gives a good visual picture of the relationship between the two variables, and aids the interpretation of the correlation coefficient or regression model. Scatter plots are similar to line graphs in that they use horizontal and vertical axes to plot data points. However, they have a very specific purpose. Scatter plots show how much one variable is affected by another. The relationship between two variables is called their correlation. 23

24 More about Scatter Plot What is a trend line? A line on a graph showing the general direction that a group of points seem to be heading. A scatter plot describes a positive trend if, as one set of values increases, the other set tends to increase. A scatter plot describes a negative trend if, as one set of values increases, the other set tends to decrease. A scatter plot shows no trend if the ordered pairs show no correlation. 24

25 Patterns of Data in Scatterplots Scatterplots are used to analyze patterns in bivariate data. These patterns are described in terms of linearity, slope, and strength. Linearity refers to whether a data pattern is linear (straight) or nonlinear (curved). Slope refers to the direction of change in variable Y when variable X gets bigger. If variable Y also gets bigger, the slope is positive; but if variable Y gets smaller, the slope is negative. Strength refers to the degree of "scatter" in the plot. If the dots are widely spread, the relationship between variables is weak. If the dots are concentrated around a line, the relationship is strong. 25

26 Patterns of Data in Scatterplots 26

The relation is strong. II. The slope is positive. III.

27 Problem The scatterplot below shows the relation between two variables. Which of the following statements are true? I. The relation is strong. II. The slope is positive. III. The slope is negative. (A) I only (B) II only (C) III only (D) I and II (E) I and III

Summarizing and Displaying Categorical Data

Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency