Introduction To Statical Theory : Data Presentation (Solution)

LEARNING OUTCOMES

At the end of this module, you will be able to:

• produce and interpret frequency distribution tables;
• produce and interpret graphs
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  FREQUENCY DISTRIBUTION:

  Data can be presented in various forms depending on the type of data collected. A frequency distribution is a table showing how often each value (or set of values) of the variable in question occurs in a data set. A frequency table is used to summarize categorical or numerical data. Frequencies are also presented as relative frequencies, that is, the percentage of the total number in the sample.
  

  EXAMPLE: Frequency distribution of peptic ulcer according to site of ulcer

  Site of ulcer	Frequency	Percent
  Gastric ulcer	24	30.0
  Duodenal ulcer	50	62.5
  Gastric and duodenal ulcer	6	7.5
  TOTAL	80	100

  GRAPHICAL METHODS:

  Frequency distributions and are usually illustrated graphically by plotting various types of graphs:
  

  Bar graph - A bar graph is a way of summarizing a set of categorical data. It displays the data using a number of rectangles, of the same width, each of which represents a particular category. Bar graphs can be displayed horizontally or vertically and they are usually drawn with a gap between the bars (rectangles).
  Histogram - A histogram is a way of summarizing data that are measured on an interval scale (either discrete or continuous). It is often used in exploratory data analysis to illustrate the features of the distribution of the data in a convenient form.
  Pie chart - A pie chart is used to display 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.
  Line graph - A line graph is particularly useful when we want to show the trend of a variable over time. Time is displayed on the horizontal axis (x-axis) and the variable is displayed on the vertical axis (y- axis).

  

  DESCRIPTIVE MEASURES:

  Measures of central tendency and dispersion are common descriptive measures for summarising numerical data.
  

  1. Measures of central tendency:

  Measures of central tendency are measures of the location of the middle or the center of a distribution.
  The most frequently used measures of central tendency are the mean, median and mode.
  
• The mean is obtained by summing the values of all the observations and dividing by the number of observations.
  
• The median (also referred to as the 50th percentile) is the middle value in a sample of ordered values. Half the values are above the median and half are below the median.
• The mode is a value occurring most frequently. It is rarely of any practical use for numerical data.

A comparison of the mean, median and mode can reveal information about skewness, as illustrated in figure below. The mean, median and mode are similar when the distribution is symmetrical. When the distribution is skewed the median is more appropriate as a measure of central tendency.

2. Measures of Dispersion:

A measure of dispersion is a numerical value describing the amount of variability present in a data set.
The standard deviation (SD) is the most commonly used measure of dispersion. With the SD you can measure dispersion relative to the scatter of the values about their mean.

The range can also be used to describe the variability in a set of data and is defined as the difference between the maximum and minimum values. The range is an appropriate measure of dispersion when the distribution is skewed.

Example:

Consider the following measurements: 8, 9, 11, 5, 12, 17, 7, 23, 39, 15 .
a) Calculate the measures of central tendency: Mean, Median and Mode:
The mean is 14.6
To locate the median, data values are ordered: 5, 7, 8, 9, 11, 12, 15, 17, 23, 39. The median lies between the 5th and 6th value = 11.5.
Mode: Not applicable, since each value occures only once.
b) Calculate the measures of dispersion: Range and Standard Deviation
Range: 5 to 39.
Standard deviation: 10.09

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How to use the calculator:

1. Click on calculator: (Alternately, if you have a Microsoft-compatible keyboard, press the "Calculator" button on the keyboard.)
2. You may have to set the calculator to scientific mode by clicking on the VIEW tab as it might be set to standard mode.
3. Then do the following to enter the data in order to compute the mean and the standard deviation:
   1. Enter your first data value.
   2. Click Sta, and then click Dat.
   3. Enter the rest of the data, clicking Dat after each entry.
   4. Click the Ave button to calculate the mean, click the s button to calculate the standard deviation.