SSC CGL Paper 3 | Tier 2 JSO Study Material | SSC CGL Tier 2 Preparation | Day 3

SSC CGL Paper 3 | Tier 2 JSO Study Material | SSC CGL Tier 2 Preparation

SSC CGL Paper 3 | Tier 2 JSO Study Material | SSC CGL Tier 2 Preparation

Dear Students,

Here, we are providing study material of Paper 3 [JSO] for SSC CGL tier 2. These  are also covering statistics syllabus For Statistical Investigator Grade-II and Assistant Audit officer (for Junior Statistical Officer (JSO), Ministry of Statistics & Programme Implementation) syllabus for cgl 2018-19.

You can also see the SSC CGL  Maths, Reasoning, G.K. and English Questions with detailed solution on our website. All the candidates who are preparing for SSC CGL 2018 must start their preparation according to the syllabus and exam pattern.

We wish you all good luck for the upcoming SSC Exams.

 

MEASURES OF CENTRAL TENDENCY

Introduction

In statistics, a central tendency (or, more commonly, a measure of central tendency) is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.

The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote “the tendency of quantitative data to cluster around some central value.”

The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysts may judge whether data has a strong or a weak central tendency based on its dispersion.

The following may be applied to one-dimensional data.

Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed.

  1. Arithmetic mean (or simply, mean) – the sum of all measurements divided by the number of observations in the data set.

  2. Median – the middle value that separates the higher half from the lower half of the data set. The median and the mode are the only measures of central tendency that can be used for ordinal data, in which values are ranked relative to each other but are not measured absolutely.
  3. Mode – the most frequent value in the data set. This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments.
  4. Geometric mean – the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale.

  5. Harmonic mean – the reciprocal of the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale.
  6. Weighted mean – an arithmetic mean that incorporates weighting to certain data elements.
  7. Truncated mean (or trimmed mean) – the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
  8. Interquartile mean – a truncated mean based on data within the interquartile range.
  9. Midrange – the arithmetic mean of the maximum and minimum values of a data set.
  10. Midhinge – the arithmetic mean of the two quartiles.
  11. Trimeanthe weighted arithmetic mean of the median and two quartiles.
  12. Winsorized mean – an arithmetic mean in which extreme values are replaced by values closer to the median.

Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. In addition, there is the

  1. Geometric median – which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.

The Quadratic mean (often known as the root mean square) is useful in engineering, but is not often used in statistics. This is because it is not a good indicator of the center of the distribution when the distribution includes negative values.

SSC CGL Paper 3

Definition

A measure of central tendency is a single value that describes the way in which a group of data cluster around a central value. To put in other words, it is a way to describe the center of a data set. There are three measures of central tendency: the mean, the median, and the mode.

According to Prof Bowley “Measures of central tendency (averages) are statistical constants which enable us to comprehend in a single effort the significance of the whole.”

Why Is Central Tendency Important?

Central tendency is very useful in psychology. It lets us know what is normal or ‘average’ for a set of data. It also condenses the data set down to one representative value, which is useful when you are working with large amounts of data. Could you imagine how difficult it would be to describe the central location of a 1,000 item data set if you had to consider every number individually?

Central tendency also allows you to compare one data set to another. For example, let’s say you have a sample of girls and a sample of boys, and you are interested in comparing their heights. By calculating the average height for each sample, you could easily draw comparisons between the girls and boys.

Central tendency is also useful when you want to compare one piece of data to the entire data set. Let’s say you received a 60% on your last psychology quiz, which is usually in the D range. You go around and talk to your classmates and find out that the average score on the quiz was 43%. In this instance, your score was significantly higher than those of your classmates. Since your teacher grades on a curve, your 60% becomes an A. Had you not known about the measures of central tendency, you probably would have been really upset by your grade and assume that you bombed the test.

SSC CGL Tier 2 | Paper 3 [JSO] Study Material Day 3

In this lesson, we will study some common measures of central tendency, viz.

(i) Arithmetical average, also called mean

(ii) Median

(iii) Mode

Objectives

After studying this lesson, you will be able to

  • define mean of raw/ungrouped and grouped data;
  • calculate mean of raw/ungrouped data and also of grouped data by ordinary and short-cut-methods;
  • define median and mode of raw/ungrouped data;
  • calculate median and mode of raw/ungrouped data.

Types of Measures Of Central Tendency

The types of measures of central tendency-

(i) Arithmetical average, also called mean

(ii) Median

(iii) Mode

SSC CGL Tier 2 | Paper 3 [JSO] Study Material Day 3

The brief description are :-

Requisites of a Good Measure of Central Tendency:

  1. It should be rigidly defined.
  2. Central Tendency should be simple to understand & easy to calculate.
  3. It should be based upon all values of given data.
  4. It should be capable of further mathematical treatment.
  5. Central Tendency should have sampling stability.
  6. It should be not be unduly affected by extreme values.

Mean

To find the arithmetic mean, add the values of all terms and them divide sum by the number of terms, the quotient is the arithmetic mean. There are three methods to find the mean :

(i) Direct method: In individual series of observations x1, x2,… xn the arithmetic mean is obtained by following formula.

c

(ii) Short-cut method: This method is used to make the calculations simpler.

Let A be any assumed mean (or any assumed number), d the deviation of the arithmetic mean, then we have

d

(iii)Step deviation method: If in a frequency table the class intervals have equal width, say I than it is convenient to use the following formula.

e

where u=(x-A)/i ,and i is length of the interval, A is the assumed mean.

Example- Compute the arithmetic mean of the following by direct and short -cut methods both:

Class           20-30   30-40   40-50    50-60      60-70

Frequency     8          26          30       20             16

SSC CGL Tier 2 | Paper 3 [JSO] Study Material Day 3

By direct method

M = (∑fx)/N = 4600/100 = 46.

By short cut method.

Let assumed mean A= 45.

M = A + (∑ fd )/N = 45+100/100 = 46.

Merits of Mean

  1. It is rigidly defined.
  2. It is easy to understand & easy to calculate.
  3. Mean is based upon all values of the given data.
  4. It is capable of further mathematical treatment.
  5. It is not much affected by sampling fluctuations.

Demerits of Mean

  1. Mean cannot be calculated if any observations are missing.
  2. It cannot be calculated for the data with open end classes.
  3. It is affected by extreme values.
  4. Mean cannot be located graphically.
  5. It may be number which is not present in the data.
  6. It can be calculated for the data representing qualitative characteristic

MEDIAN

The median is defined as the measure of the central term, when the given terms (i.e., values of the variate) are arranged in the ascending or descending order of magnitudes. In other words the median is value of the variate for which total of the frequencies above this value is equal to the total of the frequencies below this value.

Due to Corner, ―The median is the value of the variable which divides the group into two equal parts one part comprising all values greater, and the other all values less then the median‖.

For example. The marks obtained, by seven students in a paper of Statistics are 15, 20, 23, 32, 34, 39, 48 the maximum marks being 50, then the median is 32 since it is the value of the 4th term, which is situated such that the marks of 1st, 2nd and 3rd students are less than this value and those of 5th, 6th and 7th students are greater then this value.

COMPUTATION OF MEDIAN

(a) Median in individual series.

Let n be the number of values of a variate (i.e. total of all frequencies). First of all we write the values of the variate (i.e., the terms) in ascending or descending order of magnitudes

Here two cases arise:

Case 1. If n is odd then value of (n+1)/2th term gives the median.

Case2. If n is even then there are two central terms i.e., n/2th and 1  the mean of these two values gives the median.

(b) Median in continuous series (or grouped series). In this case, the median (Md) is computed by the following formula

g

Example 1 – According to the census of 1991, following are the population figure, in thousands, of 10 cities :

1400, 1250, 1670, 1800, 700, 650, 570, 488, 2100, 1700.

Find the median.

Solution. Arranging the terms in ascending order.

488, 570, 650, 700, 1250, 1400, 1670, 1800, 2100.

Here n=10, therefore the median is the mean of the measure of the 5th and 6th terms.

Here 5th term is 1250 and 6th term is 1400.

Median (Md) = (1250+14000)/2 Thousands

= 1325 Thousands

Merits of Median

  1. It is rigidly defined.
  2. Median is easy to understand & easy to calculate.
  3. It is not affected by extreme values.
  4. Even if extreme values are not known median can be calculated.
  5. It can be located just by inspection in many cases.
  6. It can be located graphically.
  7. Median is not much affected by sampling fluctuations.
  8. It can be calculated for data based on ordinal scale.

Demerits of Median

  1. It is not based upon all values of the given data.
  2. For larger data size the arrangement of data in the increasing order is difficult process.
  3. It is not capable of further mathematical treatment.
  4. It is insensitive to some changes in the data values.

Mode

The mode is the most frequent data value. Mode is the value of the variable which is predominant in the given data series. Thus in case of discrete frequency distribution, mode is the value corresponding to maximum frequency. Sometimes there may be no single mode if no one value appears more than any other. There may also be two modes (bimodal), three modes (trimodal), or more than three modes (multi-modal).

For grouped frequency distributions, the modal class is the class with the largest frequency. After identifying modal class mode is evaluated by using interpolated formula. This formula is applicable when classes are of equal width.

h

Where l1= lower limit of the modal class,

l2= upper limit of the modal class‟

d1 =fm-f0 and d2=fm-f1

Where fm= frequency of the modal class,

f0 = frequency of the class preceding to the modal class,

f1= frequency of the class succeeding to the modal class.

Mode can be located graphically by drawing histogram.

Steps:

1) Draw histogram

2) Locate modal class (highest bar of the histogram

3) Join diagonally the upper end points of the end points of the highest bar to the adjacent bars.

4) Mark the point of intersection of the diagonals.

5) Draw the perpendicular from this point on the X-axis .

6) The point where the perpendicular meets X-axis gives the modal value.

Example. Find the mode from the following size of shoes

Size of shoes      1     2   3    4     5    6    7     8      9

Frequency         1     1    1    1    2     3   2     1      1

Here maximum frequency is 3 whose term value is 6. Hence the mode is modal size number

6.

(b) In continuous frequency distribution the computation of mode is done by the following

Formula :-

h

l = lower limit of class,

f1 = frequency of modal class,

f0 =frequency of the class just preceding to the modal class,

f2 =frequency of the class just following of the modal class,

i =class interval

Merits of Mode

  1. It is easy to understand & easy to calculate.
  2. It’s not affected by extreme values or sampling fluctuations.
  3. Mode can be located just by inspection in many cases.
  4. It is always present within the data.
  5. Even if extreme values are not known mode can be calculated.
  6. It can be located graphically.
  7. It is applicable for both qualitative and quantitative data.

Demerits of Mode

  1. It’s not rigidly defined.
  2. It is not based upon all values of the given data.
  3. It is not capable of further mathematical treatment.

 

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