Histograms
Histograms are graphical representations of the distribution of numerical data. They consist of a series of adjacent rectangles, where the area of each rectangle corresponds to the frequency of data within a certain range. Histograms are commonly used to visualize the shape, center, and spread of a dataset, making them a valuable tool for data analysis.
8 Key excerpts on "Histograms"
- Bruce B. Frey(Author)
- 2018(Publication Date)
- SAGE Publications, Inc(Publisher)
...Alon Friedman Alon Friedman Friedman, Alon Histograms Histograms 780 782 Histograms A histogram is a bar chart in which data values are grouped together and put into different classes. These classes often present the frequency distributions found in the data set. Histogram coordinate systems are based on the horizontal axis and vertical axis. These two axes give the histogram the widths of the group that are equal to the class intervals and heights equal to the corresponding frequencies. Bars often represent the visual aspect of Histograms; the height of each bar corresponds to its class frequency. As a result, the histogram makes the middle of the distribution visually apparent. Histograms based on relative frequencies show the proportion of scores in each interval rather than the number of scores. This entry discusses the history of Histograms and how they are used with statistics, data, and probability distributions. History Histograms were introduced into the context of statistics as a columnar representation of frequency distributions arranged along the x axis. Karl Pearson defined Histograms as an estimate of the probability distribution of a given variable by depicting the frequencies of observations occurring in certain ranges of values, also known as continuous variables. The graphic display of a histogram is an important aspect of measuring the distribution. Although the graphic display of the histogram can show many visual patterns, many agree that a histogram should always display information succinctly. Charles Joseph Minard created an influential histogram showing the losses suffered by Napoleon’s army in the Russian campaign of 1812 (see Figure 1)...
eBook - ePubStatistics
The Essentials for Research
- Henry E. Klugh(Author)
- 2013(Publication Date)
- Psychology Press(Publisher)
...These are also plotted from the data of Table 2.2. The histogram consists of a series of adjacent bars whose heights represent the number of subjects obtaining a score and whose location on the abscissa represents the value of the score. Notice that the vertical lines marking off the bars do not originate from the center of the score interval but from its edges. The edges of the individual bars mark the theoretical limits of the score intervals along the abscissa. Figure 2.4 Histogram of the examination scores tallied in Table 2.2. Figure 2.4a A histogram of the examination scores tallied in Table 2.2. Sometimes frequency polygons, or Histograms of two different distributions, will both be plotted on the same set of coordinates. If the differences between the distributions are subtle, this procedure may highlight them. Whether one uses a frequency polygon or a histogram to represent data is largely a matter of personal preference. 2.7 Bar Charts Bar charts are the preferred graphs when data are discrete, that is, when they result from the process of counting. This convention is somewhat fluid in psychology, where ordinal scales are concerned, but it should be followed without exception for nominally scaled data, that is, for nonorderable countables. The bar chart is very much like the histogram except that spaces are left between the bars in the bar chart. Bar charts sometimes use the vertical axis to represent categories and the horizontal axis to represent frequency of occurrence. Study the bar chart in Figure 2.5 where we have graphed the enrollment in introductory courses for science departmennts at a typical college. Figure 2.5 A bar chart of enrollment in introductory science courses. 2.8 Grouped Frequency Distributions We now consider a more complex kind of frequency distribution called a grouped frequency distribution, but first we call your attention to the approximate nature of all continuous measurements...
eBook - ePub- Derek L. Waller(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...What they needed was a quick, clear, neat, and precise visual presentation of the hotel’s performance. Anthony pondered over the most appropriate presentation: an absolute frequency histogram; a relative frequency histogram; a polygon; an ogive; a stem-and-leaf display; a box and whisker plot; a line graph … ? The purpose of this Chapter 2 is to demonstrate some ways to visually present data so that the correct message gets transmitted. Chapter subjects ✓ Frequency distributions • Frequency distribution table • Absolute frequency histogram • Relative frequency histogram • Frequency polygons • Ogives • Stem-and-leaf display ✓ Visuals of quartiles and percentiles • Box and whisker plot • The percentile histogram ✓ Line graphs • Candid presentations • Geometric mean as a line graph ✓ Visuals of categorical data • Pie chart • Categorical histogram • Bar chart • Contingency tables • Stacked Histograms • Pareto diagram • Spider web diagram • Gap. analysis • Pictograms Using Microsoft Excel’s graphic capabilities, data can be transposed into visual displays that make interpretation and subsequent decision making easier. Most of us, even with the best education, cannot quickly and meaningfully interpret data such as presented in the Icebreaker. However, presented as a graph, histogram, or bar chart information can more easily be understood. All media publications: The Economist; The New York Times, Wall Street Journal; Time; Fortune and others either in a print version or on the Web use visual displays. They are relatively easy to understand. Frequency distributions Frequency distributions are groupings of data values into class limits to see if a pattern exists. The first step is to develop the group in a table and then from this plot the various visual aids. Frequency distribution table A frequency distribution table groups dataset values into unique categories according to how often, i.e. the frequency, that grouped data appear in a given category...
eBook - ePub- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...It shows the percentage for each class. It helps to understand the concept of probability and to compare 2 or more sets of data. Example : Form the frequency distribution for the following data given weights in pounds of 30 college students: Take class intervals of 10 units each. Also, construct the cumulative frequency and relative frequency distribution. Given the length of the class interval = 10 Minimum weight = 104 ; Maximum weight = 168 Range = 166 − 104 = 62 Number of intervals = range / length = 62 / 10 = 6.2 = 7 approximately Weight (lbs.) Tally Marks Frequency Cumulative Frequency Relative Frequency 100–110 \\ 2 2 2/30 = 0.067 110–120 \\\ 3 2 + 3 = 5 3/30 =. 0.100 120–130 \\\\ 5 5 + 5 = 10 5/30 = 0.167 130–140 \\\\ \\\\ 10 10 + 10 = 20 10/30 = 0.333 140–150 \\\\ \ 6 20 + 6 = 26 6/30 = 0.200 150–160 \\\ 3 26 + 3 = 29 3/30 = 0.100 160–170 \ 1 29 + 1 = 30 1/30 = 0.033 Total 30 1.000 The cumulative frequency column helps find how many elements are up to that class interval. In the preceding example, there are 5 elements up to 120 pounds. The relative frequency column helps to find the percentage of elements present in that interval. In the example, 20% items fall in the interval (150–160) pounds. 3.12 Diagrammatic Representation of Data Usually the statistical data can be presented in the form of statements and tables; additionally, it can also be represented in the form of diagrams. Diagrams are ideal visual methods of presenting data. It helps people understand the data easily. Engineers, managers, technical men, and businessmen have been using them for a long time. What is hidden in a mass of data is brought out clearly and within a second, we get a cross-sectional diagram of the whole situation. A diagram of daily or weekly sales tells the manager quickly the trend of the business. Government sectors also use the diagram to show the nation’s economic development...
eBook - ePub- Debbie L. Hahs-Vaughn, Richard Lomax(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...As you can see, along the X axis we plot the values of the variable X and along the Y axis the frequencies for each interval. The height of the bar again corresponds to the frequencies for a particular value of X. This figure represents an ungrouped histogram as the interval size is 1. That is, along the X axis the midpoint of each bar is the midpoint of the interval ; each bar begins on the left at the lower real limit of the interval, the bar ends on the right at the upper real limit, and the bar is 1 unit wide. If we wanted to use an interval size of 2, for example, using the grouped frequency distribution in Table 2.3, then we could construct a grouped histogram in the same way; the differences would be that the bars would be 2 units wide, and the height of the bars would obviously change. Try this one on your own for practice. FIGURE 2.2 Histogram of statistics quiz data. One could also plot relative frequencies on the Y axis to reflect the percentage of students in the sample whose scores fell into a particular interval. In reality, all that we have to change is the scale of the Y axis. The height of the bars would remain the same regardless of plotting frequencies or relative frequencies. For this particular dataset, each frequency corresponds to a relative frequency of.04. 2.2.3 Frequency Polygon (Line Graph) Another graphical method appropriate for data that have at least some rank order (i.e., ordinal, interval, or ratio) is the frequency polygon (i.e., line graph). A polygon is a many-sided figure. The frequency polygon is set up in a fashion similar to the histogram. However, rather than plotting a bar for each interval, points are plotted for each interval and then connected together as shown in Figure 2.3 (generated in SPSS using the default options). The X and Y axes are the same as with the histogram...
eBook - ePub- Richard Startup, Elwyn T. Whittaker(Authors)
- 2021(Publication Date)
- Routledge(Publisher)
...It is very easy to see where the two districts differ most in terms of the distribution of household size. Discrete data can very often be directly displayed in the form of a frequency table, but there are times when this is not appropriate. For example, if we consider the variable of school size as indicated by pupil numbers, the individual values that this variable can take will be over a considerable range, so that a frequency table would not summarise them at all well. In such cases, sets of values are grouped together in the form of (say) schools with numbers of pupils from 200 to 299, from 300 to 399, and so on. We would then have a grouped frequency table, as in Table 2.4. Although such a table is informative, it possesses one major drawback compared with an ordinary frequency table. The sizes of the individual schools have been irretrievably lost. This feature will pose problems when, later in this chapter, we come to analyse tables of this kind. Figure 2.3 Relative frequency bar chart. To present our grouped frequency distribution graphically, use is made of a histogram (Figure 2.4). In a case like this where all the group intervals are of equal length, we proceed to construct the histogram by representing the measurements or observations constituting the set of data (in this case pupil numbers) on a horizontal scale and the group frequencies on the vertical scale. The graph is then formed by drawing rectangles, the bases of which are supplied by the group intervals and the heights of which are given by the corresponding group frequencies. It should be noted that for ease of presentation the rectangles of Figure 2.4 have been made to meet at the lower limits of the group intervals, i.e. 300, 400, etc., even though the group upper limits, i.e. 299, 399, etc., in fact differ from the adjacent lower limits (precisely because of the underlying discrete nature of the data)...
eBook - ePub- Hugh Coolican(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...The horizontal axis might carry the mean scores obtained in several trials of an experiment. Line charts are conventionally used to demonstrate the interaction effect in a two-way ANOVA analysis (see Figure 23.3 as an example). Here we can clearly see the way that ratings on one independent variable (interviewee ethnicity/accent) vary in different ways for the two levels of the other independent variable (rater’s ethnicity). Figure 14.5 Line chart – proportion of positive affect ratings (blue line) and negative affect ratings (grey line) above criterion at each sampling time. Error bars are standard error of the mean. Source: Steptoe et al. (2008). © 2008 The British Psychological Society. Figure 14.6 Time-series chart – casualties (UK) before and after the 1967 breathalyser crackdown Source: Ross, Campbell and Glass (1973). The histogram A histogram is a way of showing the pattern of the whole data set to a reader. It communicates information about the shape of the distribution of values found. The extroversion scores data from Table 14.1 are depicted in a histogram in Figure 14.7. Key Term Histogram Chart containing whole of continuous data set divided into class intervals with each interval represented by a column proportional to frequency in the interval. Note that each category (or ‘bin’) of the frequency table is represented by one vertical bar. Frequency is usually shown on the y- (vertical) axis. The scale or class intervals are shown on the X-Axis. All the bars are joined because the chart represents a continuous and whole group of scores, and therefore a gap must be left where a category is empty (see the column for 17) to show there are no scores in that class interval. Each column is the same width, and since the height of each column represents the number of values found in that category, it follows that the area of each column is proportional to the number of cases it represents throughout the histogram...
eBook - ePub- Todd L. VanPool, Robert D. Leonard(Authors)
- 2011(Publication Date)
- Wiley-Blackwell(Publisher)
...A histogram looks similar to tally marks from a frequency table laid on their side. Let us use an example of data gathered in the Gallina area of New Mexico to illustrate the histogram. Table 3.2 presents 100 measurements of minimum sherd thickness measured to examine differences in ceramic technology across settlements in the project area. These measurements were taken using 0.5 mm increments. Thus, the value 4.5 mm reflects a sherd with a minimum thickness between 4.25 and 4.75 mm. Table 3.2 Minimum ceramic sherd thickness (mm) from the Gallina region of New Mexico Again, patterns in the raw data are difficult to identify. To make sense of our data, we must first organize them. Figure 3.5 presents a frequency distribution of these data. Once a frequency distribution is constructed, it is quite easy to construct a histogram – a visual that is slightly more aesthetically pleasing (and publishable) than the tallies, lines, and numbers in Figure 3.5. Figure 3.6 is a histogram of the data presented in Table 3.2. Figure 3.5 Frequency distribution of the Gallina ceramic minimum sherd thicknesses (mm) Figure 3.6 Histogram of minimum sherd thicknesses For Figure 3.6, the metric scale of measurement is depicted on the horizontal axis of the graph. The number of variates in each class is counted, and the counts of variates are depicted using vertical bars called bins. For example, there are 35 variates with a minimum sherd width of 5.0 mm. The bin associated with the class of 5.0 mm consequently rises to the level associated with a frequency of 35 variates. This histogram provides an excellent visual presentation of the shape, or distribution, of the data. We can see that the distribution is roughly symmetrical, and that values toward the center are more common than extreme values...
