This type of chart is widely used for nominal and ordinal qualitative variables, but it can also be used for discrete quantitative variables, because it allows us to investigate the presence of data trends.

As its name indicates, through bars, this chart represents the absolute or relative frequencies of each possible category (or numeric value) of a qualitative variable (or quantitative). In vertical bar charts, each variable category is shown on the X-axis as a bar with constant width, and the height of the respective bar indicates the frequency of the category on the Y-axis. Conversely, in horizontal bar charts, each variable category is shown on the Y-axis as a bar of constant height, and the length of the respective bar indicates the frequency of the category on the X-axis.

Let’s now build horizontal and vertical bar charts from a practical example.

Example 3.4

A bank created a satisfaction survey, which was used with 120 customers, trying to measure how agile its services were (excellent, good, satisfactory, and poor). The absolute frequencies for each category are presented in Table 3.E.8. Construct a vertical and horizontal bar chart for this problem.

Table 3.E.8

Frequencies of Occurrences per Category

Satisfaction	Absolute Frequency
Excellent	58
Good	18
Satisfactory	32
Poor	12

Solution

Let’s build the vertical and horizontal bar charts of Example 3.4 in Excel.

First, the data in Table 3.E.8 must be standardized, codified, and selected in a spreadsheet. After that, we can click on the Insert tab and, in the Charts group, and select the option Columns. The chart is automatically generated on the screen.

Next, to personalize the chart, while clicking on it, we must select the following icons on the Layout tab: (a) Axis Titles: let’s select the title for the horizontal axis (Satisfaction) and for the vertical axis (Frequency); (b) Legend: to hide the legend, we must click on None; (c) Data Labels: clicking on More Data Label Options, the option Value must be selected in Label Contains (or we can select the option Outside End).

Fig. 3.2 shows the vertical bar chart of Example 3.4 generated in Excel.

Based on Fig. 3.2, we can see that the categories of the variable being analyzed are presented on the X-axis by bars with the same width and their respective heights indicate the frequencies on the Y-axis.

To construct the horizontal bar chart, we must select the option Bar instead of Columns. The other steps follow the same logic. Fig. 3.3 represents the frequency data from Table 3.E.8 through a horizontal bar chart constructed in Excel.

The horizontal bar chart in Fig. 3.3 represents the categories of the variable on the Y-axis and their respective frequencies on the X-axis. For each variable category, we draw a bar with a length that corresponds to its frequency.

Therefore, this chart only offers information related to the behavior of each category of the original variable and to the generation of investigations regarding the type of distribution, not allowing us to calculate position, dispersion, skewness or kurtosis measures, since the variable being studied is qualitative.

3.3.1.2 Pie Chart

Another way to represent qualitative data, in terms of relative frequencies (percentages), is the definition of pie charts. The chart corresponds to a circle with a random radius (the whole) divided into sectors or slices of pie of several different sizes (parts of the whole).

This chart allows the researcher to visualize the data as slices of a pie or parts of a whole. Let’s now build the pie chart from a practical example.

Example 3.5

An election poll was carried out in the city of Sao Paulo to check voters’ preferences concerning the political parties running in the next elections for Mayor. The percentage of voters per political party can be seen in Table 3.E.9. Construct a pie chart for Example 3.5.

Table 3.E.9

Percentage of Voters per Political Party

Political Party	Percentage
PMDB	18
PSDB	22
PDT	12.5
PT	24.5
PC do B	8
PV	5
Others	10

Solution

Let’s build the pie chart for Example 3.5 in Excel. The steps are similar to the ones in Example 3.4. However, we now have to select the option Pie in the Charts group, on the Insert tab. Fig. 3.4 presents the pie chart obtained in Excel for the data shown in Table 3.E.9.

3.3.1.3 Pareto Chart

The Pareto chart is a Quality control tool and has as its main objective to investigate the types of problems and, consequently, to identify their respective causes, so that an action can be taken in order to reduce or eliminate them.

The Pareto chart is a chart that contains bars and a line graph. The bars represent the absolute frequencies of occurrences of problems and the lines represent the relative cumulative frequencies. The problems are sorted in descending order of priority. Let’s now illustrate a practical example with a Pareto chart.

Example 3.6

A manufacturer of credit and magnetic cards has as its main objective to reduce the number of defective cards. The quality inspector classified a sample of 1000 cards that were collected during one week of production, according to the types of defects found, as shown in Table 3.E.10. Construct a Pareto chart for this problem.

Table 3.E.10

Frequencies of the Occurrence of Each Defect

Type of Defect	Absolute Frequency (Fi)
Damaged/Bent	71
Perforated	28
Illegible printing	12
Wrong characters	20
Wrong numbers	44
Others	6
Total	181

Solution

The first step in generating a Pareto chart is to sort the defects in order of priority (from the highest to the lowest frequency). The bar chart represents the absolute frequency of each defect. To construct the line graph, it is necessary to calculate the relative cumulative frequency (%) up to the defect analyzed. Table 3.E.11 shows the absolute frequency for each type of defect, in descending order, and the relative cumulative frequency (%).

Table 3.E.11

Absolute Frequency for Each Defect and the Relative Cumulative Frequency (%)

Type of Defect	Number of Defects	Cumulative %
Damaged/Bent	71	39.23
Wrong numbers	44	63.54
Perforated	28	79.01
Wrong characters	20	90.06
Illegible printing	12	96.69
Others	6	100

Let’s now build a Pareto chart for Example 3.6 in Excel, using the data in Table 3.E.11.

First, the data in Table 3.E.11 must be standardized, codified, and selected in an Excel spreadsheet. In the Charts group, on the Insert tab, let’s select the option Columns (and the clustered column subtype). Note that the chart is automatically generated on the screen. However, absolute frequency data as well as relative cumulative frequency data are presented as columns. To change the type of chart related to the cumulative percentage, we must click with the right button on any bar of the respective series and select the option Change Series Chart Type, followed by a line graph with markers. The resulting chart is a Pareto chart.

To personalize the Pareto chart, we must use the following icons on the Layout tab: (a) Axis Titles: for the bar chart, we selected the title for the horizontal axis (Type of defect) and for the vertical axis (Frequency); for the line graph, we called the vertical axis Percentage; (b) Legend: to hide the legend, we must click on None; (c) Data Table: let’s select the option Show Data Table with Legend Keys; (d) Axes: the main unit of the vertical axes for both charts is set in 20 and the maximum value of the vertical axis for line graphs, in 100.

Fig. 3.5 shows the chart constructed in Excel that corresponds to the Pareto chart for Example 3.6.

In a line graph, points are represented by the intersection of the variables involved on the horizontal axis (X) and on the vertical axis (Y), and they are connected by straight lines.

Despite considering two axes, line graphs will be used in this chapter to represent the behavior of a single variable. The graph shows the evolution or trend of a quantitative variable’s data, which is usually continuous, at regular intervals. The numeric variable values are represented on the Y-axis, and the X-axis only shows the data distribution in a uniform way. Let’s now illustrate a practical example of a line graph.

Example 3.7

Cheap & Easy is a supermarket that registered the percentage of losses it had in the last 12 months (Table 3.E.12). After having done that, it will adopt new prevention measures. Build a line graph for Example 3.7.

Table 3.E.12

Percentage of Losses in the Last 12 Months

Month	Losses (%)
January	0.42
February	0.38
March	0.12
April	0.34
May	0.22
June	0.15
July	0.18
August	0.31
September	0.47
October	0.24
November	0.42
December	0.09

Solution

To build the line graph for Example 3.7 in Excel, in the Charts group, on the Insert tab, we must select the option Lines. The other steps follow the same logic of the previous examples. The complete chart can be seen in Fig. 3.6.

3.3.2.2 Scatter Plot

A scatter plot is very similar to a line graph. The biggest difference between them is in the way the data are plotted on the horizontal axis.

Similar to a line graph, here the points are also represented by the intersection of the variables along the X-axis and the vertical axis. However, they are not connected by straight lines.

The scatter plot studied in this chapter is used to show the evolution or trend of a single quantitative variable’s data, similar to the line graph; however, at irregular intervals (in general). Analogous to a line graph, the numeric variable values are represented on the Y-axis and the X-axis only represents the data behavior throughout time.

In the next chapter, we will see how a scatter plot can be used to describe the behavior of two variables simultaneously (bivariate analysis). The numeric values of one variable will be represented on the Y-axis and the other one on the X-axis.

Example 3.8

Papermisto is the supplier of three types of raw materials for the production of paper: cellulose, mechanical pulp, and trimmings. In order to maintain its quality standards, the factory carries out a rigorous inspection of its products during each production phase. At irregular intervals, an operator must verify the esthetic and dimensional characteristics of the product selected with specialized instruments. For instance, in the cellulose storage phase, the product must be piled up in bales of approximately 250 kg each. Table 3.E.13 shows the weight of the bales collected in the last 5 hours, at irregular intervals, varying between 20 and 45 minutes. Construct a scatter plot for Example 3.8.

Table 3.E.13

Evolution of the Weight of the Bales Throughout Time

Time (min)	Weight (kg)
30	250
50	255
85	252
106	248
138	250
178	249
198	252
222	251
252	250
297	245

Solution

To build the scatter plot for Example 3.8 in Excel, in the Charts group, on the Insert tab, we must select the option Scatter. The other steps follow the same logic of the previous examples. The scatter plot can be seen in Fig. 3.7.

3.3.2.3 Histogram

A histogram is a vertical bar chart that represents the frequency distribution of one quantitative variable (discrete or continuous). The variable values being studied are presented on the X-axis (the base of each bar, with a constant width, represents each possible value of the discrete variable or each class of continuous values, sorted in ascending order). On the other hand, the height of the bars on the Y-axis represents the frequency distribution (absolute, relative, or cumulative) of the respective variable values.

A histogram is very similar to a Pareto chart. It is also one of the seven quality tools. A Pareto chart represents the frequency distribution of a qualitative variable (types of problem), whose categories represented on the X-axis are sorted in order of priority (from the category with the highest frequency to the one with the lowest). A histogram represents the frequency distribution of a quantitative variable, whose values represented on the X-axis are sorted in ascending order.

Therefore, the first step to elaborate a histogram is building the frequency distribution table. As presented in Sections 3.2.2 and 3.2.3, for each possible value of a discrete variable or for a class with continuous data, we calculate the absolute frequency, the relative frequency, the cumulative frequency, and the relative cumulative frequency. The data must be sorted in ascending order.

The histogram is then constructed from this table. The first column of the frequency distribution table, which represents the numeric values or the classes with the values of the variable being studied, will be presented on the X-axis, and the column of absolute frequency (or relative frequency, cumulative frequency, or relative cumulative frequency) will be presented on the Y-axis.

Many pieces of statistical software generate the histogram automatically, from the original values of the quantitative variable being studied, without having to calculate the frequencies. Even though Excel has the option of building a histogram from analysis tools, we will show how to build it from the column chart, due to its simplicity.

Example 3.9

In order to improve their services, a national bank is hiring new managers to serve their corporate clients. Table 3.E.14 shows the number of companies dealt with daily in one of their main branches in the capital. Elaborate a histogram from these data using Excel.

Table 3.E.14

Number of Companies Dealt With Daily

13	11	13	10	11	12	8	12	9	10
12	10	8	11	9	11	14	11	10	9

Solution

The first step is building the frequency distribution table:

From the data in Table 3.E.15, we can build a histogram of absolute frequency, relative frequency, cumulative frequency, or relative cumulative frequency using Excel. The histogram generated will be the absolute frequency one.

Thus, we must standardize, codify, and select the first two columns of Table 3.E.15 (except the last row: Sum) in an Excel spreadsheet. In the Charts group, on the Insert tab, let’s select the option Columns.

Let’s click on the chart so that it can be personalized. On the Layout tab, we selected the following icons: (a) Axis Titles: select the title for the horizontal axis (Number of companies) and for the vertical axis (Absolute frequency); (b) Legend: to hide the legend, we must click on None. The histogram generated in Excel can be seen in Fig. 3.8.

Table 3.E.15

Frequency Distribution for Example 3.9

Number of Companies	Fi	Fri (%)	Fac	Frac (%)
8	2	10	2	10
9	3	15	5	25
10	4	20	9	45
11	5	25	14	70
12	3	15	17	85
13	2	10	19	95
14	1	5	20	100
Sum	20	100

As mentioned, many statistical computer packages, including SPSS and Stata, build the histogram automatically from the original data of the variable being studied (in this example, using the data in Table 3.E.14), without having to calculate the frequencies. Moreover, these packages have the option of plotting the normal curve.

Fig. 3.9 shows the histogram generated using SPSS (with the option of a normal curve) using the data in Table 3.E.14. We will see this in detail in Sections 3.6 and 3.7, how it can be constructed using SPSS and Stata software, respectively.

Note that the values of the discrete variable are presented in the middle of the base.

For continuous variables, consider the data in Table 3.E.5 (Example 3.3), regarding the grades of the students enrolled in the subject Financial Market. These data were sorted in ascending order, as presented in Table 3.E.6.

Fig. 3.10 shows the histogram generated using SPSS software (with the option of a normal curve) using the data in Table 3.E.5 or Table 3.E.6.

Note that the data were grouped considering an interval between h = 0.5 classes, differently from Example 3.3 that considered h = 1. The classes’ lower limits are represented on the left side of the base of the bar, and the upper limits (not included in the class) on the right side. The height of the bar represents the total frequency in the class. For example, the first bar represents the 3.5 ├ 4.0 class and there are three values in this interval (3.5, 3.8 and 3.9).

3.3.2.4 Stem-and-Leaf Plot

Both bar charts and histograms represent the shape of the variable’s frequency distribution. The stem-and-leaf plot is an alternative to represent the frequency distributions of discrete and continuous quantitative variables with few observations, with the advantage of maintaining the original value of each observation (it allows the visualization of all data information).

In the plot, the representation of each observation is divided into two parts, separated by a vertical line: the stem is located on the left of the vertical line and represents the observation’s first digit(s); the leaf is located on the right of the vertical line and represents the observation’s last digit(s). Choosing the number of initial digits that will form the stem or the number of complementary digits that will form the leaf is random. The stems usually contain the most significant digits, and the leaves the least significant.

The stems are represented in a single column and their different values throughout many lines. For each stem represented on the left-hand side of the vertical line, we have the respective leaves shown on the right-hand side throughout many columns. Stems as well as leaves must be sorted in ascending order. In the cases in which there are too many leaves per stem, we can have more than one line with the same stem. Choosing the number of lines is random, as well as defining the interval or the number of classes in a frequency distribution.

To build a stem-and-leaf plot, we can follow the sequence of steps:

Step 1: Sort the data in ascending order, to make the visualization of the data easier.

Step 2: Define the number of initial digits that will form the stem, or the number of complementary digits that will form the leaf.

Step 3: Elaborate the stems, represented in a single column on the left of the vertical line. Their different values are represented throughout many lines, in ascending order. When the number of leaves by stem is very high, we can define two or more lines for the same stem.

Step 4: Place the leaves that correspond to the respective stems, on the right-hand side of the vertical line, throughout many columns (in ascending order).

Example 3.10

A small company collected its employees’ ages, as shown in Table 3.E.16. Build a stem-and-leaf plot.

Table 3.E.16

Employees’ Ages

44	60	22	49	31	58	42	63	33	37
54	55	40	71	55	62	35	45	59	54
50	51	24	31	40	73	28	35	75	48

Solution

To construct the stem-and-leaf plot, let’s apply the four steps described:

• Step 1

First, we must sort the data in ascending order, as shown in Table 3.E.17.

• Step 2

Table 3.E.17

Employees’ Ages in Ascending Order

22	24	28	31	31	33	35	35	37	40
40	42	44	45	48	49	50	51	54	54
55	55	58	59	60	62	63	71	73	75

The next step to construct a stem-and-leaf plot is to define the number of initial digits of the observation that will form the stem. The complementary digits will form the leaf. In this example, all of the observations have two digits. The stems correspond to the tens and the leaves correspond to the units.

• Step 3

The following step is to build the stems. Based on Table 3.E.17, we can see that there are observations that begin with the tens 2, 3, 4, 5, 6, and 7 (stems). The stem with the highest frequency is 5 (8 observations), it is possible to represent all of its leaves in a single line. Therefore, we will have a single line per stem. Hence, the stems are presented in a single column on the left of the vertical line, in ascending order, as shown in Fig. 3.11.

• Step 4

Finally, let’s place the leaves that correspond to each stem on the right-hand side of the vertical line. The leaves are represented in ascending order throughout many columns. For example, stem 2 contains leaves 2, 4, and 8. Stem 5 contains leaves 0, 1, 4, 4, 5, 5, 8, and 9, represented throughout 8 columns. If this stem were divided into two lines, the first line would have leaves 0 to 4, and the second line leaves 5 to 9.

Fig. 3.12 illustrates the stem-and-leaf plot for Example 3.10.

Example 3.11

The average temperature, in Celsius, registered in the last 40 days in the city of Porto Alegre can be found in Table 3.E.18. Elaborate the stem-and-leaf plot for Example 3.11.

Table 3.E.18

Average Temperature in Celsius

8.5	13.7	12.9	9.4	11.7	19.2	12.8	9.7	19.5	11.5
15.5	16.0	20.4	17.4	18.0	14.4	14.8	13.0	16.6	20.2
17.9	17.7	16.9	15.2	18.5	17.8	16.2	16.4	18.2	16.9
18.7	19.6	13.2	17.2	20.5	14.1	16.1	15.9	18.8	15.7

Solution

Once again, let’s apply the four steps to construct the stem-and-leaf plot, but now we have to consider continuous variables.

• Step 1

First, let’s sort the data in ascending order, as shown in Table 3.E.19.

• Step 2

Table 3.E.19

Average Temperature in Ascending Order

8.5	9.4	9.7	11.5	11.7	12.8	12.9	13.0	13.2	13.7
14.1	14.4	14.8	15.2	15.5	15.7	15.9	16.0	16.1	16.2
16.4	16.6	16.9	16.9	17.2	17.4	17.7	17.8	17.9	18.0
18.2	18.5	18.7	18.8	19.2	19.5	19.6	20.2	20.4	20.5

In this example, the leaves correspond to the last digit. The remaining digits (to the left) correspond to the stems.

• Steps 3 and 4

The stems vary from 8 to 20. The stem with the highest frequency is 16 (7 observations), and its leaves can be represented in a single line. For each stem, we place the respective leaves. Fig. 3.13 shows the stem-and-leaf plot for Example 3.11.

3.3.2.5 Boxplot or Box-and-Whisker Diagram

The boxplot (or box-and-whisker diagram) is a graphical representation of five measures of position or location of a certain variable: minimum value, first quartile (Q₁), second quartile (Q₂) or median (Md), third quartile (Q₃) and maximum value. From a sorted sample, the median corresponds to the central position and the quartiles to subdivisions of the sample, four equal parts, each one containing 25% of the data.

Thus, the first quartile (Q₁) describes 25% of the first data (organized in ascending order). The second quartile corresponds to the median (50% of the sorted data are located below it and the remaining 50% above it), and the third quartile (Q₁₃) corresponds to 75% of the observations. The dispersion measure resulting from these location measures is called interquartile range (IQR) or interquartile interval (IQI) and corresponds to the difference between Q₃ and Q₁.

This plot allows us to assess the data symmetry and distribution. It also gives us a visual perspective of whether or not there are discrepant data (univariate outliers), since these data are above the upper and lower limits. A representation of the diagram can be seen in Fig. 3.14.

Calculating the median, the first, and third quartiles, and investigating the existence of univariate outliers will be discussed in Sections 3.4.1.1, 3.4.1.2, and 3.4.1.3, respectively. In Sections 3.6.3 and 3.7, we will study how to generate the box-and-whisker diagram on SPSS and Stata, respectively, using a practical example.

3.4.1.1 Measures of Central Tendency

The most common measures of central tendency are the arithmetic mean, the median, and the mode.

3.4.1.2 Quantiles

According to Bussab and Morettin (2011), only the use of measures of central tendency may not be suitable to represent a set of data, since they are also impacted by extreme values. Moreover, only with the use of these measures, it is not possible for the researcher to have a clear idea of the data dispersion and symmetry. As an alternative, we can use quantiles, such as, quartiles, deciles, and percentiles. The 2nd quartile (Q₂), 5th decile (D₅), or 50th percentile (P₅₀) correspond to the median; therefore, they are measures of central tendency.

3.4.1.2.1 Quartiles

Quartiles (Q_i, i = 1, 2, 3) are measures of position that divide a set of data into four parts with equal dimensions, sorted in ascending order.

Thus, the 1st Quartile (Q₁ or the 25th percentile) indicates that 25% of the data are less than Q₁, or that 75% of the data are greater than Q₁.

The 2nd Quartile (Q₂, or the 5th decile, or the 50th percentile) corresponds to the median, indicating that 50% of the data are less or greater than Q₂.

The 3rd Quartile (Q₃ or the 75th percentile) indicates that 75% of the data are less than Q₃, or that 25% of the data are greater than Q₃.

3.4.1.2.2 Deciles

Deciles (D_i, _i = 1, 2, ..., 9) are measures of position that divide a set of data into 10 equal parts, sorted in ascending order.

Therefore, the 1st decile (D₁ or 10th percentile) indicates that 10% of the data are less than D₁ or that 90% of the data are greater than D₁.

The 2nd decile (D₂ or 20th percentile) indicates that 20% of the data are less than D₂ or that 80% of the data are greater than D₂.

And so on, and so forth, until the 9th decile (D₉ or 90th percentile), indicating that 90% of the data are less than D₉ or that 10% of the data are greater than D₉.

3.4.1.3 Identifying the Existence of Univariate Outliers

A dataset can contain observations that are extremely distant from most observations or that are inconsistent. These observations are called outliers or atypical, discrepant, abnormal, or extreme values.

Before deciding what will be done with the outliers, we must know the causes that lead to such an occurrence. In many cases, these causes can determine the most suitable treatment for the respective outliers. The main causes are measurement mistakes, execution/implementation mistakes, and variability inherent to the population.

There are many outlier identification methods: boxplots, discordance models, Dixon’s test, Grubbs’ test, Z-scores, among others. In the Appendix of Chapter 11 (Cluster Analysis), a very efficient method for detecting multivariate outliers will be presented (BACON algorithm—Blocked Adaptive Computationally Efficient Outlier Nominators).

The existence of outliers through boxplots (the construction of boxplots was studied in Section 3.3.2.5) is identified from the IQR (interquartile range), which corresponds to the difference between the third and first quartiles:

$\mathrm{IQR}=Q_3-Q_1$

Note that the IQR is the length of the box. Any values located below Q₁ or above Q₃ by 1.5 ∙ IQR more will be considered mild outliers and will be represented by circles. They may even be accepted in the population, but with some suspicion. Thus, the X° value of a variable is considered a mild outlier when:

$X^{°}<Q_1\mathit{-}1.5\bullet \mathrm{IQR}$

$X^{°}>Q_3\mathit{+}1.5\bullet \mathrm{IQR}$

or any values located below Q₁ or above Q₃ by 3 ∙ IQR more will be considered extreme outliers and will be presented by asterisks. Thus, the X⁎ value of a variable is considered an extreme outlier when:

$X^{\ast }<Q_1-3.\mathrm{IQR}$

$X^{\ast }>Q_3+3.\mathrm{IQR}$

Fig. 3.15 illustrates the boxplot with the identification of outliers.

3.4.2.1 Range

The simplest measure of variability is the total range, or simply range (R), which represents the difference between the highest and lowest value of the set of data:

$R=X_{\max }-X_{\min }$

3.4.2.2 Average Deviation

Deviation is the difference between each observed value and the mean of the variable. Thus, for population data, it would be represented by (X_i − μ), and for sample data, by $\left(X_i-\overline{X}\right)$. The modulus or absolute deviation ignores the ± sign and is denoted by $\left|X_i-\overline{X}\right|$.

Average deviation, or absolute average deviation, represents the arithmetic mean of absolute deviations.

3.4.2.2.2 Case 2: Average Deviation of Grouped Discrete Data

For grouped data, presented in a frequency distribution table for m groups, the calculation of the average deviation is:

$\overline{D}=\frac{\sum _{i=1}^m\left|X_i-\mu \right|.F_i}{N}\left(,\mathbf{for},,\mathbf{the},,\mathbf{population}\right)$

  (3.30)

$\overline{D}=\frac{\sum _{i=1}^m\left|X_i-\overline{X}\right|.F_i}{n}\left(,\mathbf{for},,\mathbf{samples}\right)$

  (3.31)

bearing in mind that $\overline{X}=\frac{\sum _{i=1}^m{X}_i.F_i}{n}$.

Example 3.29

Table 3.E.33 shows the number of goals scored by the D.C. soccer team in their last 30 games, with their respective absolute frequencies. Calculate the average deviation.

Table 3.E.33

Frequency Distribution of Example 3.29

Number of Goals	Fi
0	5
1	8
2	6
3	4
4	4
5	2
6	1
Sum	30

Solution

The mean is $\overline{X}=\frac{0\times 5+1\times 8+\cdots +6\times 1}{30}=2.133$. The average deviation can be determined from the calculations presented in Table 3.E.34:

Table 3.E.34

Calculations of the Average Deviation for Example 3.29

Number of Goals	Fi	$\left|X_i-\overline{X}\right|$	$\left|X_i-\overline{X}\right|.F_i$
0	5	2.133	10.667
1	8	1.133	9.067
2	6	0.133	0.800
3	4	0.867	3.467
4	4	1.867	7.467
5	2	2.867	5.733
6	1	3.867	3.867
Sum	30		41.067

Therefore, $\overline{D}=\frac{\sum _{i=1}^m\left|X_i-\overline{X}\right|.F_i}{n}=\frac{41.067}{30}=1.369$.

3.4.2.2.3 Case 3: Average Deviation of Continuous Data Grouped into Classes

For continuous data grouped into classes, the calculation of the average deviation is:

$\overline{D}=\frac{\sum _{i=1}^k\left|X_i-\mu \right|.F_i}{N}\left(,\mathbf{for},,\mathbf{the},,\mathbf{population}\right)$

  (3.32)

$\overline{D}=\frac{\sum _{i=1}^k\left|X_i-\overline{X}\right|.F_i}{n}\left(,\mathbf{for},,\mathbf{samples}\right)$

  (3.33)

Note that Expressions (3.32) and (3.33) are similar to Expressions (3.30) and (3.31), respectively, except that, instead of m groups, we consider k classes. Moreover, X_i represents the middle or central point of each class i, where $\overline{X}=\frac{\sum _{i=1}^k{X}_i.F_i}{n}$, as presented in Expression (3.6).

Example 3.30

In order to determine its variation due to genetic factors, a survey with 100 newborn babies collected information about their weight. Table 3.E.35 shows the data grouped into classes and their respective absolute frequencies. Calculate the average deviation.

Table 3.E.35

Newborn Babies’ Weight (in kg) Grouped into Classes

Class	Fi
2.0 ├ 2.5	10
2.5 ├ 3.0	24
3.0 ├ 3.5	31
3.5 ├ 4.0	22
4.0 ├ 4.5	13
Sum

Solution

First, we must calculate $\overline{X}$:

$\overline{X}=\frac{\sum _{i=1}^k{X}_i.F_i}{n}=\frac{2.25\times 10+2.75\times 24+3.25\times 31+3.75\times 22+4.25\times 13}{100}=3.270$

The average deviation can be determined from the calculations presented in Table 3.E.36:

Table 3.E.36

Calculations of the Average Deviation for Example 3.30

Class	Fi	Xi	$\left|X_i-\overline{X}\right|$	$\left|X_i-\overline{X}\right|.F_i$
2.0 ├ 2.5	10	2.25	1.02	10.20
2.5 ├ 3.0	24	2.75	0.52	12.48
3.0 ├ 3.5	31	3.25	0.02	0.62
3.5 ├ 4.0	22	3.75	0.48	10.56
4.0 ├ 4.5	13	4.25	0.98	12.74
Sum	100			46.6

Therefore, $\overline{D}=\frac{\sum _{i=1}^k\left|X_i-\overline{X}\right|.F_i}{n}=\frac{46.6}{100}=0.466$.

3.4.2.3 Variance

Variance is a measure of dispersion or variability that evaluates how much the data are dispersed in relation to the arithmetic mean. Thus, the higher the variance, the higher the data dispersion.

3.4.2.3.2 Case 2: Variance of Grouped Discrete Data

For grouped data, represented in a frequency distribution table by m groups, the variance can be calculated as follows:

${\sigma }^2=\frac{\sum _{i=1}^m{\left(X_i-\mu \right)}^2.F_i}{N}=\frac{\sum _{i=1}^m{X}_i^2.F_i-\frac{{\left(\sum _{i=1}^m{X}_i.F_i\right)}^2}{N}}{N}\left(,\mathbf{for},,\mathbf{the},,\mathbf{population}\right)$

  (3.37)

$S^2=\frac{\sum _{i=1}^m{\left(X_i-\overline{X}\right)}^2.F_i}{n-1}=\frac{\sum _{i=1}^m{X}_i^2.F_i-\frac{{\left(\sum _{i=1}^m{X}_i.F_i\right)}^2}{n}}{n-1}\left(,\mathbf{for},,\mathbf{samples}\right)$

  (3.38)

where $\overline{X}=\frac{\sum _{i=1}^m{X}_i.F_i}{n}$.

Example 3.32

Consider the data in Example 3.29 regarding the number of goals scored by the D.C. soccer team in the last 30 games, with their respective absolute frequencies. Calculate the variance.

Solution

As calculated in Example 3.29, the mean is $\overline{X}=2.133$. The variance can be determined from the calculations presented in Table 3.E.37:

Table 3.E.37

Calculations of the Variance

Number of Goals	Fi	${\left(X_i-\overline{X}\right)}^2$	${\left(X_i-\overline{X}\right)}^2.F_i$
0	5	4.551	22.756
1	8	1.284	10.276
2	6	0.018	0.107
3	4	0.751	3.004
4	4	3.484	13.938
5	2	8.218	16.436
6	1	14.951	14.951
Sum	30		81.467

Therefore, $S^2=\frac{\sum _{i=1}^m{\left(X_i-\overline{X}\right)}^2.F_i}{n-1}=\frac{81.467}{29}=2.809$

3.4.2.3.3 Case 3: Variance of Continuous Data Grouped into Classes

For continuous data grouped into classes, we calculate the variance as follows:

${\sigma }^2=\frac{\sum _{i=1}^k{\left(X_i-\mu \right)}^2.F_i}{N}=\frac{\sum _{i=1}^k{X}_i^2.F_i-\frac{{\left(\sum _{i=1}^k{X}_i.F_i\right)}^2}{N}}{N}\left(,\mathbf{for},,\mathbf{the},,\mathbf{population}\right)$

  (3.39)

$S^2=\frac{\sum _{i=1}^k{\left(X_i-\overline{x}\right)}^2.F_i}{n-1}=\frac{\sum _{i=1}^k{X}_i^2.F_i-\frac{{\left(\sum _{i=1}^k{X}_i.F_i\right)}^2}{n}}{n-1}\left(,\mathbf{for},,\mathbf{samples}\right)$

  (3.40)

Note that Expressions (3.39) and (3.40) are similar to Expressions (3.37) and (3.38), respectively, except that we consider k classes instead of m groups.

Example 3.33

Consider the data in Example 3.30 regarding the weight of newborn babies grouped into classes, with their respective absolute frequencies. Calculate the variance.

Solution

As calculated in Example 3.30, we have $\overline{X}=3.270$.

The variance can be determined from the calculations presented in Table 3.E.38:

Table 3.E.38

Calculations of the Variance for Example 3.33

Class	Fi	Xi	${\left(X_i-\overline{X}\right)}^2$	${\left(X_i-\overline{X}\right)}^2.F_i$
2.0 ├ 2.5	10	2.25	1.0404	10.404
2.5 ├ 3.0	24	2.75	0.2704	6.4896
3.0 ├ 3.5	31	3.25	0.0004	0.0124
3.5 ├ 4.0	22	3.75	0.2304	5.0688
4.0 ├ 4.5	13	4.25	0.9604	12.4852
Sum	100			34.46

Therefore, $S^2=\frac{\sum _{i=1}^k{\left(X_i-\overline{X}\right)}^2.F_i}{n-1}=\frac{34.46}{99}=0.348$.

3.4.2.4 Standard Deviation

Since the variance considers the mean of squared deviations, its value tends to be very high and difficult to interpret. To solve this problem, we calculate the square root of the variance. This measure is known as the standard deviation. It is calculated as follows:

$\sigma =\sqrt{{\sigma }^2}\left(,\mathbf{for},,\mathbf{the},,\mathbf{population}\right)$

$S=\sqrt{S^2}\left(,\mathbf{for},,\mathbf{samples}\right)$

3.4.2.5 Standard Error

The standard error is the standard deviation of the mean. It is obtained by dividing the standard deviation by the square root of the population or sample size:

${\sigma }_{\overline{X}}=\frac{\sigma }{\sqrt{N}}\mathbf{for}\mathbf{the}\mathbf{population}$

$S_{\overline{X}}=\frac{S}{\sqrt{n}}\mathbf{for}\mathbf{samples}$

The higher the number of measurements, the better the determination of the average value will be (higher accuracy), due to the compensation of random errors.