Question: What is a measure of central tendency and what are the common measures of central tendency? Also, when is the median preferred over the mean?

The most common measure of central tendency is the mode (i.e., the most frequently occurring number)

The median (i.e., the middle point or fiftieth percentile), and the mean (i.e., the arithmetic average).

The median is preferred over the mean when the numbers are highly skewed (i.e., non-normally distributed).

Since, measures of central tendency condense a bunch of information into a single, digestible value that represents the center of the data, this makes measures of central tendencies important for several reasons:

Summarizing data: Instead of listing every data point, one can use a central tendency measure to get a quick idea of what is typical in the data set.

Comparisons: By computing central tendency measures for different groups or datasets, one can easily compare them to see if there are any differences.

Decision making: Central tendency measures can help to make wise decisions. For instance, knowing the average income in an area can help set prices. Imagine an organization is analyzing customer purchases. Knowing the average amount spent can help them tailor promotions or target specific customer groups.

Identifying trends: Measures of central tendencies may help in observing the trend over time. This can be done by using different visualizations to see if there are any trends, like a rise in average house prices.

However, it is very important to understand these Measures of Central Tendency (mean, median, mode). Each measure of central tendency has its strengths and weaknesses. Choosing the right measure of central tendency depends on the kind of data and what one’s interest is to extract from and try to understand.

A frequency table is a way of summarizing a set of data. It is a record of each value (or set of values) of the variable in the data/question. In this post, we will learn about the ways of Constructing Frequency Tables for discrete and continuous data.

A grouping of qualitative data into mutually exclusive classes showing the number of observations in each class is called a 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 variables are arranged into different classes with their frequencies, then this is known as continuous frequency distribution. If data of discrete variables is arranged into different classes with their frequencies then it is known as discrete distribution or discontinuous distribution.

Discrete Frequency Distribution Table Example

Car Type

Number of Cars

Local

50

Foreign

30

Total Cars

80

Constructing Frequency Tables

Constructing Frequency tables (distributions) may be done for both discrete and continuous variables. A discrete frequency distribution can be converted back to original values, but for continuous variables, it is not possible.

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

Calculate the range of the data. The range is the difference between the highest and smallest values of the given data. \[Range = Highest Value – Lowest Value\]

Decide the number of Classes. The 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.

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 \[I=\frac{Highest Value – Lowest Value}{Number of Classes}=\frac{H-L}{K}\] Where $I$ is the class interval, $H$ is the highest observed value, $L$ 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 for certain situations such as to avoid a large number of empty or almost empty classes.

Set the Individual Class Limits Class limits are the endpoints in the class interval. State clear class limits so that you can put each of the observations into one and only one category i.e. you must avoid overlapping or unclear class limits. Class intervals are usually rounded up to get a convenient class size, and cover a larger than necessary range. It is convenient to choose the endpoints of the class interval so that no observation falls on them. It can be obtained by expressing the endpoints to one more place of decimal than the observations themselves, i.e. limits are converted to class boundaries to achieve continuity in data.

Tally the Observation into the Classes

Count the Number of Items in each Class The number of observations in each class I called the class frequency. Note that totaling the frequencies in each class must equal 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.

Note: Arranging/organizing the data into a tabulation or frequency distribution results in a loss of detailed information as the 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 in constructing the frequency tables, the advantages of condensing the data into a more understandable and organized form are more than offset this disadvantage.

The post is about “The Word Statistics Meaning and Use”.

The word statistics was first used by German scholar Gottfried Achenwall in the middle of the 18^{th} 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 the 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 was used by E.A.W. Zimmermann in “A Political Survey of the Present State of Europe”. It appeared in the 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 and also interpretation of data through appropriate analysis.

The Word Statistics Now a Day

Now statistics are being used with 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 as a plural i.e. a set of numerical information. For instance statistics on 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 numerical data to make inferences and to reach an appropriate decision in a 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 the decision on numerical data.

The word statistics refers to numerical quantities calculated from sample observations; a single quantity calculated from sample observations is called statistics such as the 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 is singular sense methods.

In another way, the word Statistics has two meanings:

The science of data: In this sense, statistics deals with collecting, analyzing, interpreting, and presenting numerical data. Therefore, statistics helps us to understand the world around us by making sense of large amounts of information. Statisticians use a variety of techniques to summarize data, identify patterns, and draw wise conclusions.

Pieces of data: Statistics also refers to the actual numerical data itself, for example, averages, percentages, or other findings from a study. The real-life examples of statistics are: (i) unemployment statistics or (ii) crime statistics.

For learning about the Basics of Statistics Follow the link Basic Statistics

Skewness is the lack of symmetry (lack of normality) in a probability distribution. The skewness is usually quantified by the index as given below

$$s = \frac{\mu_3}{\mu_2^{3/2}}$$

where $\mu_2$ and $\mu_3$ are the second and third moments about the mean.

Table of Contents

This index formula described above takes the value zero for a symmetrical distribution. A distribution is positively skewed when it has a longer and thin tail to the right. A distribution is negatively skewed when it has a longer thin tail to the left.

Any distribution is said to be skewed when the data points cluster more toward one side of the scale than the other. Creating such a curve that is not symmetrical.

Skewed Data

The two general rules for Skewed Data are

If the mean is less than the median, the data are skewed to the left, and

If the mean is greater than the median, the data are skewed to the right.

Therefore, if the mean is much greater than the median the data are probably skewed to the right.

Misinterpretation of Mean and Median: The mean can be sensitive to outliers in skewed distributions and might not accurately represent the “typical” value. The median, which is the middle value when the data is ordered, can be a more robust measure of the central tendency for skewed data.

Statistical Tests: Some statistical tests assume normality (zero skewness). If the data is skewed, alternative tests or transformations might be necessary for reliable results.

Identifying Skewed Data

There are a couple of ways to identify skewed data:

Visual Inspection: Histograms and box plots are useful tools for visualizing the distribution of the data. Skewed distributions will show an asymmetry in the plots.

Skewness Coefficient: This statistic measures the direction and magnitude of the skew in the distribution. A positive value indicates a positive skew, a negative value indicates a negative skew, and zero indicates a symmetrical distribution.