Month: April 2020

• It is the process of arranging data into homogeneous (similar) groups according to their common characteristics.
• Raw data cannot be easily understood and it is not fit for further analysis and interpretation. This arrangement of data helps users in comparison and analysis.
• For example, the Population of town can be grouped according to sex, age, marital status etc.

Classification of data

The method of arranging data into homogeneous classes according to some common features present in the data is called classification.

A planned data analysis system makes fundamental data easy to find and recover. This can be of particular interest for legal discovery, risk management and compliance. Written methods and set of guidelines for data classification should determine what levels and measures the company will use to organise data and define the roles of employees within the business regarding input stewardship. Once a data-classification scheme has been designed, security standards that stipulate proper approaching practices for each division and storage criteria that determine the data’s lifecycle demands should be discussed.

Objectives of Data Classification

The primary objectives of data classification are:

• To consolidate the volume of data in such a way that similarities and differences can be quickly understood. Figures can consequently be ordered in a few sections holding common traits.
• To aid comparison.
• To point out the important characteristics of the data at a flash.
• To give importance to the prominent data collected while separating the optional elements.
• To allow a statistical method of the material gathered.

Methods of Classification

An important branch of mathematics that deals with gathering, organizing, estimating and interpreting the vast numerical data for a survey or a research, is known as statistics. There may be one or more numbers of statistical data that are used more than once. The number of times a particular data item is utilized, is known as its frequency.

When the distribution of frequencies is listed in a table OR tabular presentation of frequency distribution, known as frequency table. It is used to list out one or more variables taken in a sample. Each sample contains an individual frequency and each frequency is distributed with an interval between each frequency. It is also of two types that is univariate and joint. Frequency distribution can be defined as a summary presentation of the number of observations of an attribute or values of a variable arranged according to their magnitudes either individually in the case of discrete series or in a range or class interval in the case of both discrete and continuing series.

Frequency Table

Frequency Distribution Table is a way to organize data. A frequency distribution table is an organized tabulation of the number of individual events located in each category. It contains at least two columns, one for the score categories (X) and another for the frequencies (f). Below we have explained briefly for you to understand the concept of frequency table better and workout frequency table example:

Solved Example

Question: Here is the list of marks obtained for the students in the examination. Find the number of students who got more than 85 marks, More than 95, Less than 80 more than 76.

Solution:

From the table we can conclude that:

Students who got more than 85 = 8 + 5 + 1 = 14

Students who got more than 95 = 1

Students who got less than 80 more than 76 = 14.

Construction of Frequency Distribution

The following steps are involved in the construction of a frequency distribution.

(1) Find the range of the data: The range is the difference between the largest and the smallest values.

(2) Decide the approximate number of classes in which the data are to be grouped. There are no hard and first rules for number of classes. In most cases we have 5 to 20 classes. H.A. Sturges provides a formula for determining the approximation number of classes.

K=1+3.322logN

where K= Number of classes

and logN = Logarithm of the total number of observations.

Example: If the total number of observations is 50, the number of classes would be

K=1+3.322logN

K=1+3.322log50

K=1+3.322(1.69897)

K=1+5.644

K=6.644

7 classes, approximately.

(3) Determine the approximate class interval size: The size of class interval is obtained by dividing the range of data by the number of classes and is denoted by h class interval size

h = Range Number of Classes

In the case of fractional results, the next higher whole number is taken as the size of the class interval.

(4) Decide the starting point: The lower class limit or class boundary should cover the smallest value in the raw data. It is a multiple of class intervals.

Example: 0,5,10,15,20, etc. are commonly used.

(5) Determine the remaining class limits (boundary): When the lowest class boundary has been decided, by adding the class interval size to the lower class boundary you can compute the upper class boundary. The remaining lower and upper class limits may be determined by adding the class interval size repeatedly till the largest value of the data is observed in the class.

(6) Distribute the data into respective classes: All the observations are divided into respective classes by using the tally bar (tally mark) method, which is suitable for tabulating the observations into respective classes. The number of tally bars is counted to get the frequency against each class. The frequency of all the classes is noted to get the grouped data or frequency distribution of the data. The total of the frequency columns must be equal to the number of observations.

Data can be presented in the form of organized information, combined in tables or even graphically represented. Imagine seeing a set of data in the written form or in tabular form versus a graph that gives you the same information. Isn’t it simpler and quicker to comprehend data if we can visually see it?

It is for this purpose that data can be organized graphically for interpretation in a single glance in Statistics. The two forms of graphical representation that we shall cover in this lesson are bar diagram and histogram.

Bar Diagram

Also known as a column graph, a bar graph or a bar diagram is a pictorial representation of data. It is shown in the form of rectangles spaced out with equal spaces between them and having equal width. The equal width and equal space criteria are important characteristics of a bar graph.

Note that the height (or length) of each bar corresponds to the frequency of a particular observation. You can draw bar graphs both, vertically or horizontally depending on whether you take the frequency along the vertical or horizontal axes respectively. Let us take an example to understand how a bar graph is drawn.

The above table depicts the number of students of a class engaged in any one of the three sports given. Note that the number of students is actually the frequency. So, if we take frequency to be represented on the y-axis and the sports on the x-axis, taking each unit on the y-axis to be equal to 5 students, we would get a graph that resembles the one below.

The blue rectangles here are called bars. Note that the bars have equal width and are equally spaced, as mentioned above. This is a simple bar diagram.

Histogram

A bar diagram easy to understand but what is a histogram? Unlike a bar graph that depicts discrete data, histograms depict continuous data. The continuous data takes the form of class intervals. Thus, a histogram is a graphical representation of a frequency distribution with class intervals or attributes as the base and frequency as the height.

The key difference is that histograms have bars without any spaces between them and the rectangles need not be of equal width. So, we will understand histograms using an example.

In this case, see that we are considering class intervals such as 0-5, 5-10, 10-15 and 15-20. These are continuous data. In case, the class intervals given to you are not continuous, you must make it continuous first.

Here, you can interpret the histogram using the information that the graph gives. Consider the frequency to be as given on the left vertical axis and ignore the values on the right vertical axis. Thus, for the class interval 0-5, the corresponding frequency is 3. Again, for 5-10, the frequency is 7, and so on.

Note that we have taken the simple case of a histogram with bars of equal width. But as mentioned, it might not be the case if the class intervals are not even in size. In that case, you will get a histogram with bars stuck to each other (without any space between them) but with different widths. It could look something like this, but exactly how it will look depends on the data:

A pie chart (or a pie graph) is a circular statistical graphical chart, which is divided into slices in order to explain or illustrate numerical proportions. In a pie chart, centeral angle, area and an arc length of each slice is proportional to the quantity or percentages it represents. Total percentages should be 100 and total of the arc measures should be 360° Following illustration of pie graph depicts the cost of construction of a house.

From this graph, one can compare the sum spent on cement, steel and so on. One can also compute the actual sum spent on each individual expense. Consider an example, where we want to know how much more is the labour cost when compared to cost of steel.

Amount spent on labor =9060×600000=$ 150000

Sum spent on steel =54/360×600000=$ 90000

Excess=150000−90000=$ 60000

Let 60000=x% of 600000

⟹x/100×600000=$ 60000

⟹x=10% of total expense.

The word Ogive is a term used in architecture to describe curves or curved shapes. Ogives are graphs that are used to estimate how many numbers lie below or above a particular variable or value in data. To construct an Ogive, firstly, the cumulative frequency of the variables is calculated using a frequency table. It is done by adding the frequencies of all the previous variables in the given data set. The result or the last number in the cumulative frequency table is always equal to the total frequencies of the variables. The most commonly used graphs of the frequency distribution are histogram, frequency polygon, frequency curve, Ogives (cumulative frequency curves).

Ogives

The Ogive is defined as the frequency distribution graph of a series. The Ogive is a graph of a cumulative distribution, which explains data values on the horizontal plane axis and either the cumulative relative frequencies, the cumulative frequencies or cumulative percent frequencies on the vertical axis. Cumulative frequency is defined as the sum of all the previous frequencies up to the current point. To find the popularity of the given data or the likelihood of the data that fall within the certain frequency range, Ogive curve helps in finding those details accurately. Create the Ogive by plotting the point corresponding to the cumulative frequency of each class interval. Most of the Statisticians use Ogive curve, to illustrate the data in the pictorial representation. It helps in estimating the number of observations which are less than or equal to the particular value.

Ogive Graph

The graphs of the frequency distribution are frequency graphs that are used to exhibit the characteristics of discrete and continuous data. Such figures are more appealing to the eye than the tabulated data. It helps us to facilitate the comparative study of two or more frequency distributions. We can relate the shape and pattern of the two frequency distributions. The two methods of Ogives are

• Less than Ogive
• Greater than or more than Ogive

The graph given above represents less than and the greater than Ogive curve. The rising curve (Brown Curve) represents the less than Ogive, and the falling curve (Green Curve) represents the greater than Ogive.

Less than Ogive

The frequencies of all preceding classes are added to the frequency of a class. This series is called the less than cumulative series. It is constructed by adding the first-class frequency to the second-class frequency and then to the third class frequency and so on. The downward cumulation results in the less than cumulative series.

Greater than or More than Ogive

The frequencies of the succeeding classes are added to the frequency of a class. This series is called the more than or greater than cumulative series. It is constructed by subtracting the first class second class frequency from the total, third class frequency from that and so on. The upward cumulation result is greater than or more than the cumulative series.

Ogive Chart

An Ogive Chart is a curve of the cumulative frequency distribution or cumulative relative frequency distribution. For drawing such a curve, the frequencies must be expressed as a percentage of the total frequency. Then, such percentages are cumulated and plotted as in the case of an Ogive. Here, the steps for constructing the less than and greater than Ogive are given.

How to Draw Less Than Ogive Curve?

• Draw and mark the horizontal and vertical axes.
• Take the cumulative frequencies along the y-axis (vertical axis) and the upper-class limits on the x-axis (horizontal axis).
• Against each upper-class limit, plot the cumulative frequencies.
• Connect the points with a continuous curve.

How to Draw Greater than or More than Ogive Curve?

• Draw and mark the horizontal and vertical axes.
• Take the cumulative frequencies along the y-axis (vertical axis) and the lower-class limits on the x-axis (horizontal axis).
• Against each lower-class limit, plot the cumulative frequencies
• Connect the points with a continuous curve.

Uses of Ogive Curve

Ogive Graph or the cumulative frequency graphs are used to find the median of the given set of data. If both the less than and the greater than cumulative frequency curve is drawn on the same graph, we can easily find the median value. The point in which both the curve intersects, corresponding to the x-axis gives the median value.  Apart from finding the medians, Ogives are used in computing the percentiles of the data set values.

The median of a set of data values is the middle value of the data set when it has been arranged in ascending order.  That is, from the smallest value to the highest value.

Example:

The marks of nine students in a geography test that had a maximum possible mark of 50 are given below:

Find the median of this set of data values.

Solution:

Arrange the data values in order from the lowest value to the highest value:

The fifth data value, 36, is the middle value in this arrangement.

Merits or Uses of Median:

1. Median is rigidly defined as in the case of Mean.
2. Even if the value of extreme item is much different from other values, it is not much affected by these values e.g. Median in case of 4, 7, 12, 18, 19 is 12 and if we add two values equal to 450 10000, new median is 18.
3. It can also be used for the Quantities; those can’t give A.M; as is in case of intelligence etc. It is possible to arrange in any order and to locate the middle valve. For such cases it is the best measure.
4. It can be located graphically.
5. For open end intervals, it is also suitable one. As taking any value of the intervals, value of Median remains the same.
6. It can be easily calculated and is also easy to understand
7. Median is also used for other statistical devices such as Mean Deviation and skewness.
8. It can be located by inspection in some cases.
9. Extreme items may not be available to get Median. Only if number of terms is known, we can get median e.g.

Find median of the 9 terms, out of which first two and last three terms are missing and middle four terms are 7, 9, 10, 14. Here we can calculate as following let nine terms be

Here out of nine terms middle term is; (n+1/2) Thus 10 is the Median.

Demerits or Limitations of Median:

1. Even if the value of extreme items is too large, it does not affect too much, but due to this reason, sometimes median does not remain the representative of the series.
2. It is affected much more by fluctuations of sampling than A.M.
3. Median cannot be used for further algebraic treatment. Unlike mean we can neither find total of terms as in case of A.M. nor median of some groups when combined.
4. In a continuous series it has to be interpolated. We can find its true-value only if the frequencies are uniformly spread over the whole class interval in which median lies.
5. If the number of series is even, we can only make its estimate; as the A.M. of two middle terms is taken as Median.

Graphical Method

The median value of a series may be determinded through the graphic presentation of data in the form of Ogives.This can be done in 2 ways.

1. Presenting the data graphically in the form of ‘less than’ ogive or ‘more than’ ogive .
   2. Presenting the data graphically and simultaneously in the form of ‘less than’ and ‘more than’ ogives.The two ogives are drawn together.
2. Less than Ogive approach

Steps involved in calculating median using less than Ogive approach:
1. Convert the series into a ‘less than ‘ cumulative frequency distribution as shown above.

2. Let N be the total number of students who’s data is given.N will also be the cumulative frequency of the last interval.Find the (N/2)^th item(student) and mark it on the y-axis.In this case the (N/2)^th item (student) is 200/2 = 100^th student.
3. Draw a perpendicular from 100 to the right to cut the Ogive curve at point A.
4. From point A where the Ogive curve is cut, draw a perpendicular on the x-axis. The point at which it touches the x-axis will be the median value of the series as shown in the graph.

More than Ogive approach

Steps involved in calculating median using more than Ogive approach:
1. Convert the series into a ‘more than ‘ cumulative frequency distribution as shown above .
2. Let N be the total number of students who’s data is given.N will also be the cumulative frequency of the last interval.Find the (N/2)^th item(student) and mark it on the y-axis.In this case the (N/2)^th item (student) is 200/2 = 100^th student.
3. Draw a perpendicular from 100 to the right to cut the Ogive curve at point A.
4.From point A where the Ogive curve is cut, draw a perpendicular on the x-axis. The point at which it touches the x-axis will be the median value of the series as shown in the graph.

2. Less than and more than Ogive approach

Another way of graphical determination of median is through simultaneous graphic presentation of both the less than and more than Ogives.

1.Mark the point A where the Ogive curves cut each other.
2.Draw a perpendicular from A on the x-axis. The corresponding value on the x-axis would be the median value.

Mode is the value which occurs most frequently in a set of observations. Simply put, it is the number which is repeated most, i.e. the number with the highest frequency. In the field of statistics, it is an important tool to interpret data in a relevant manner. Now it is possible for the data set to be multimodal (have more than one mode) which means more than one observation has the same number of frequencies.

Example: Let us find the Mode of the following data

4, 89, 65, 11, 54, 11, 90, 56

Here in these varied observations the most occurring number is 11, hence the Mode = 11

Mode of Grouped Data

As we know that Mode is the most frequently occurring number of a data set. This is easily recognizable in an ungrouped dataset, but if the data set is presented in class intervals, this can get a bit tricky. So how can we calculate Mode of grouped data?

Steps to be followed to calculate the Mode are,

1. Create a table with two columns
2. In column 1 write your class intervals
3. In column 2 write the corresponding frequencies
4. Locate the maximum frequency denoted by f_m
5. Determine the class corresponding to f_m , this will be your Modal class
6. Calculate the Mode using given formula

Mode = L +fm−f1(2fm−f1−f2) × h

Where,

L = lower limit of Modal Class

f_m = frequency of modal class

h = width of modal class

f₁ = frequency of pre modal class

f₂ = frequency of post modal class

Relation between Mean, Median, and Mode

There is an inter-relation between the measures of central tendency. Professor Karl Pearson has suggested an empirical relationship between Mean, Median, and Mode. Via this equation, if the values of two measures are known we can find the third measure. The equation is as follows

Mean – Mode = 3 [Mean – Median]

Finding Mode Graphically

The following steps must be followed to find the mode graphically.

1. Represent the given data in the form of a Histogram.The hight of the rectangles in the histogram is marked by the frequencies of the class interval as shown in the graph .Identify the highest rectangle. This corresponds to the modal class of the series.
2. Join the top corners of the modal rectangle with the immediately next corners of the adjacent rectangles. The two lines must be cutting each other.This might be difficult to visualise so look at the graph given below.
3. Let the point where the joining lines cut each other be ‘A’. Draw a perpendicular line from point A onto the x-axis. The point ‘P’ where the perpendicular will meet the x-axis will give the mode.

The Histogram

In this case the value of point P turns out to be 44.12

The mean, median, and mode are all useful measures of central tendency, but their value can be limited by unique characteristics of the underlying data. A comparison across alternate measures is useful for determining the extent to which a consistent pattern of central tendency emerges. If the mean, median, and mode all coincide at a single sample observation, the sample data are said to be symmetrical. If the data are perfectly symmetrical, then the distribution of data above the mean is a perfect mirror image of the data distribution below the mean. A perfectly symmetrical distribution is illustrated in Figure. Whereas a symmetrical distribution implies balance in sample dispersion, skewness implies a lack of balance. If the greater bulk of sample observations are found to the left of the sample mean, then the sample is said to be skewed downward or to the left as in Figure. If the greater bulk of sample observations are found to the right of the mean, then the sample is said to be skewed upward or to the right as in Figure). When alternate measures of central tendency converge on a single value or narrow range of values, managers can be confident that an important characteristic of a fairly homogeneous sample of observations has been discovered. When alternate measures of central tendency fail to converge on a single value or range of values, then it is likely that underlying data comprise a heterogeneous sample of observations with important subsample differences. A comparison of alternate measures of central tendency is usually an important first step to determining whether a more detailed analysis of subsample differences is necessary.

The Mean, Median, and Mode