Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Constructing Frequency Tables

A frequency table is a way of summarizing a set of data. It is a record of the each value (or set of values) of the variable in data/question.

A grouping of qualitative data into mutually exclusive classes showing the number of observations in each class is called frequency table. The number of values falling in a particular category/class is called the frequency of that category/class denoted by f.

If data of continuous variable is arranged into different classes with their frequencies then this is known as continuous frequency distribution. If data of discrete variable is arranged into different classes with their frequencies then it is known as discrete distribution or discontinuous distribution.

Example

Car type Number of cars

Local

50

Foreign

30

Total Cars

80

Frequency distribution may be constructed both for discrete and continuous variables. Discrete frequency distribution can be converted back to original values, but for continuous variables it is not possible.

Following steps are taken into account while constructing frequency tables for continuous data.

  1. Calculate the range of the data. Range is the difference of the highest and smallest values of the given data.
    Range = Highest Value – Lowest Value
  2. Decide the number of Classes. Maximum number of classes may be determined by the formula
    Number of classes $C = 2^k$     OR    Number of classes $(C) = 1+3.3 log (n)$
    Note that: Too many classes or too few classes might not reveal the basic shape of the data set.
  3. Determine the Class Interval or Width
    The class all taken together should cover at least the distance from the lowest value in the data up to the highest value, which can be done by this formula \[I=\frac{Highest Value – Lowest Value}{Number of Classes}\]
    Where I is the class interval, H is the highest observed value, and L is the lowest observed value and K is the number of classes.
    Generally the class interval or width should be the same for all classes.
    In particular interval size is usually rounded up to some convenient number, such as a multiple of 10 or 100. Unequal class intervals present problems in graphically portraying the distribution and in doing some of the computations. Unequal class intervals may be necessary in certain situations such as to avoid a large number of empty or almost empty classes.
  4. Set the Individual Class Limits
    Class limits are the end points in class interval. State clear class limits so that you can put each of the observation into one and only one category i.e. you must avoid the overlapping or unclear class limits. Because class intervals are usually rounded up to get a convenient class size, cover a larger than necessary range.
    It is convenient to choose the end points of the class interval so that no observation falls on them. It can be obtained by expressing the end points to one more place of decimal than the observations themselves, i.e. limits are converted to class boundaries to achieve continuity in data.
  5. Tally the Observation into the Classes
  6. Count the Number of Items in each Class
    The number of observation in each class I called the class frequency. Note the totaling the frequencies in each class must equals the total number of observations. After following these steps, we have organized the data into a tabulation form which is called a frequency distribution, which can be used to summarize the pattern in the observation i.e. the concentration of the data.
Constructing Frequency Tables

Frequency Distribution Table

Note: Arranging/organizing the data into a tabulation or frequency distribution results in loss of detailed information as individuality of observations vanishes i.e. in frequency distribution we cannot pinpoint the exact value, and we cannot tell the actual lowest and highest values of the data. However the lower limit of the largest class conveys some essentially the same meaning. So the advantages of condensing the data into a more understandable and organized form are more than offset this disadvantage.

Further Reading

http://itfeature.com/statistics/frequency-distribution-table

 

The word statistics was first used by a German scholar Gotifried Achenwall

Introduction to Statistics

The word statistics was first used by a German scholar Gotifried Achenwall in the middle of the 18th century as the science of statecraft concerning the collection and use of data by the state.

The word statistics comes from the Latin word “Status” or Italian word “Statistia” or German word “Statistik” or the French word “Statistique”; meaning a political state, and originally meant information useful to the state, such as information about sizes of population (human, animal, products etc) and armed forces.

According to pioneer statistician Yule, the word statistics occurred at the earliest in the book “the element of universal erudition” by Baron (1770). In 1787 a wider definition used by E.A.W. Zimmermann in “A Political survey of the present state of Europe”. It appeared in encyclopedia of Britannica in 1797 and was used by Sir John Sinclair in Britain in a series of volumes published between 1791 and 1799 giving a statistical account of Scotland. In the 19th century, the word statistics acquired a wider meaning covering numerical data of almost any subject whatever and also interpretation of data through appropriate analysis.

Now statistics is being used in different meanings.

  • Statistics refers to “numerical facts that are arranged systematically in the form of tables or charts etc. In this sense it is always used a plural i.e. a set of numerical information. For instance statistics of prices, road accidents, crimes, births, educational institutions etc.
  • The word statistics is defined as a discipline that includes procedures and techniques used to collect, process and analyze the numerical data to make inferences and to reach appropriate decision in situation of uncertainty (uncertainty refers to incompleteness, it does not imply ignorance). In this sense word statistic is used in the singular sense. It denotes the science of basing decision on numerical data.
  • The word statistics are numerical quantities calculated from sample observations; a single quantity calculated from sample observations is called statistics such as mean. Here word statistics is plural.

“We compute statistics from statistics by statistics”

The first place of statistics is plural of statistics, in second place is plural sense data and in third place in singular sense methods.

What are the advantages of using interval estimation rather than point estimation?

Advantages of using the Interval and Point Estimation

The problem with using a point estimate is that although it is the single best guess you can make about the value of a population parameter, it is also usually wrong.

  • A major advantage of using interval estimation is that you provide a range of values with a known probability of capturing the population parameter (e.g., if you obtain from SPSS a 95% confidence interval you can claim to have 95% confidence that it will include the true population parameter.
  • An interval estimate (i.e., confidence intervals) also helps one to not be so confident that the population value is exactly equal to the single point estimate. That is, it makes us more careful in how we interpret our data and helps keep us in proper perspective.
  • Actually, perhaps the best thing of all to do is to provide both the point estimate and the interval estimate. For example, our best estimate of the population mean is the value $32,640 (the point estimate) and our 95% confidence interval is $30,913.71 to $34,366.29.
  • By the way, note that the bigger your sample size, the more narrow the confidence interval will be.
  • If you want narrow (i.e., very precise) confidence intervals, then remember to include a lot of participants in your research study.
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