## Measures Of Central Tendency – CS Foundation Statistics Notes

Central tendency is defined as “the statistical measure that identifies a single value as representative of an entire distribution.” It aims to provide an accurate description of the entire data. It is the single value that is the most typical representative of the collected data.

1. Average

‘An average is an attempt to find one single figure to describe the whole of figures’ this was the definition given by Clark and Sekkade for average. Simpson defined average as ‘A measure of central tendency is a typical value around which other figures congregate’. Average refers to the sum of numbers divided by n. Sums of data divided by the number of items in the data will give the mean average.

Average should have some characteristic features like

- It should be firmly defined. For the end-user, there should not be any confusion regarding the figure of average provided
- Average should take into consideration all the items of the series
- Even the layman should be able to understand the average provided.
- If the average has obtained two different sources for the same subject then it should not differ too much.

2. Types of average

- Mathematical average
- Positional average

1. Mathematical average

Mathematical averages cover arithmetic mean, geometric mean, and harmonic mean.

(a) Arithmetic means – The arithmetic mean of a set of data is found by taking the sum of the data, and then dividing the sum by the total number of values in the set. A mean is commonly referred to as an average. When the values are large then special methods are used so as to cut short the labor. One of these methods is called the ‘short cut method’, some middle value or if there is frequency then the largest frequency is subtracted from all the values. The values so obtained are averaged.

In case the class interval is uniform then the computation of average can be done by step deviation method. In the above three mean all the observations were equally important but there may be some situations where some observations may be more important than other observations. In such a situation, the weighted average method is used.

Example

The following frequency distribution showing the marks obtained by 50 students in statistics at a certain college.

Marks | 20-29 | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 |

Frequency | 1 | 5 | 12 | 15 | 9 | 6 | 2 |

Direct
Method |
Short-Cut Method | Step-Deviation Method | |||||

Marks | f | X | fx | D=x-A | fD | U=x-A/h | fu |

20-29 | 1 | 24.5 | 24.5 | -30 | -30 | -3 | -3 |

30-39 | 5 | 34.5 | 172.5 | -20 | -100 | -2 | -10 |

40-49 | 12 | 44.5 | 534.5 | -10 | -120 | -1 | -12 |

50-59 | 15 | 54.5 | 817.5 | 0 | 0 | 0 | 0 |

60-69 | 9 | 64.5 | 580.5 | 10 | 90 | 1 | 9 |

70-79 | 6 | 74.5 | 447.5 | 20 | 120 | 2 | 12 |

80-89 | 2 | 84.5 | 169.5 | 30 | 60 | 3 | 6 |

Total | 50 | 274.5 | 20 | 2 |

Direct Method:

274.5/50 = 54.9 or 55 marks

Short-cut method:

54.5 +20/50 = 54.5 + . 4 = 54.9

Step – Deviation Method

A = 54.5

h – 10

54.5+2/50*10 = 54.9

Properties of arithmetic

The properties are explained below with suitable illustrations.

Property 1:

If x is the arithmetic mean of n observations x_{1}, x_{2}, x_{3},.. x_{n}; then

(x_{1} – x) + (x_{2} – x) + (x_{3} – x) + … + (x_{n} – x) = 0.

Property 2:

The mean of n observations x_{1}, x_{2}, . . ., x_{n} is x. If each observation is increased by p, the mean of the new observations is (x + p).

Property 3:

The mean of n observations x_{1}, x_{2}, . . ., x_{n} is x. If each observation is decreased by p, the mean of the new observations is (x – p).

Property 4:

The mean of n observations x_{1}, x_{2}, . . ,,x_{n} is x. If each observation is multiplied by a nonzero number p, the mean of the new observations is px.

Property 5:

The mean of n observations x_{1}, x_{2}, . . ., x_{n} is x. If each observation is divided by a nonzero number p, the mean of the new observations is (x/p).

3. Merits of Arithmetic Mean

- Arithmetic Mean is simple to understand.
- Arithmetic Mean can be easily calculated.
- Arithmetic Mean can be determined in most cases.
- Arithmetic Mean is based on the observations of the series.
- Arithmetic Mean is capable of further algebraic treatment.
- Arithmetic Mean is stable. It does not differ from sample to sample.

4. Demerits of Arithmetic Mean

- Arithmetic Mean is affected by extreme values.
- Arithmetic mean need not coincide with any of the observed values.
- It is not good in the case of ratios and percentages.
- Sometimes it will give us absurd answers.

For example, the mean number of children in a family is 3.2. This can never happen. The mean number of children can be either 3 or 4 but not 3.2. Hence, the result can be misleading. - It is not suitable in the case of open-end classes. When the classes are open, we are not sure what must be the upper limit of the class.

5. Median

In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population or a probability distribution from the lower half. If you place a set of numbers in order, the median number is the middle one. If there are two middle numbers, the median is the mean of those two numbers.

6. Determination of Median

The following steps are involved in the determination of median:

(i) The given observations are arranged in either ascending or descending order of magnitude.

(ii) Given that there are n observations, the median is given by:

- The size of n+1/2 the observations, when n is odd.
- The mean of the sizes of n/2th and n+1/2 of observations, when n is even.

Consider the observations: 13, 16, 16, 17, 17, 18, 19, 21,23. On the basis of the method given above, their median is 17.

7. Characteristic features

- Since the variable, in a grouped frequency distribution, is assumed to be continuous we always take an exact value of N/2, including figures after decimals, when N is odd.
- The above formula is also applicable when classes are of unequal width.
- Median can be computed even if there are open-end classes because here we need to know only the frequencies of classes preceding or following the median class.

8. Merits of median

- Median is often used to convey the typical observation. It is primarily affected by the number of observations rather than their size.
- It is easier to calculate the median than the mean in many cases. In some cases, the median can be calculated just by inspection
- Median is very useful in the case of open-ended classes
- The Median can be measured graphically.
- Median is not affected by extreme values.
- Median serves as the most appropriate average to deal with qualitative data.
- The median value is always a certain and specific value in the series.

9. Demerits of median

- Median is incapable of further algebraic treatment.
- Though the median considers the frequency of all observations, it does so only for counting purposes and does not really consider their magnitude.
- The value of the median is affected more by sampling fluctuations than the value of the mean.
- For computing, median data needs to be arranged in ascending or descending order.
- It is not based on all the observations of the data.
- It is not accurate when the data is not large.
- In some cases, the median is determined approximately as the mid-point of two observations whereas for the mean this does not happen.

10. Mode

The mode is the value that appears most often in a set of data. For categorical or discrete variables the mode is simply the most observed value. To work out the mode, observations do not have to be placed in order, although for ease of calculation it is advisable to do so. For example in the set: 3, 3, 6, 9, 16,16,16, 27, 27, 37,52 16 is the mode since it appears more times than any other number.

A set of numbers can have more than one mode (this is known as bimodal) if there are multiple numbers that occur with equal frequency, and more times than the others in the set. 3,3,3, 9,16, 16, 16, 27, 37,48 In this example, both the number 3 and the number 16 are modes. If no number in a set of numbers occurs more than once, that set has no mode Broadly speaking, the mode is the value of the variable occurring most frequently. It is the most common value found in a series.

The mode can be calculated in three ways

(a) Just by inspection

(b) By method of grouping

(c) By interpolation formula

(a) When frequency distribution is regular then just by looking at the series we can tell which observation is maximum times in the series and that is the mode, (just by inspection)

(b) When the distribution is not regular then this method is used, (method of grouping)

Data table

Size | Frequency | Size | Frequency |

5 | 48 | 13 | 52 |

6 | 52 | 14 | 41 |

7 | 56 | 15 | 57 |

8 | 60 | 16 | 63 |

9 | 63 | 17 | 52 |

10 | 57 | 18 | 48 |

11 | 55 | 19 | 40 |

12 | 50 | – | – |

Location of mode by grouping

The frequencies in columns (1) are first added in two’s in columns (2) and (3). Then they are added in three’s in columns (4), (5), and (6). The maximum frequency in each column is indicated by thick letters. It will be observed that mode changes with the change in a grouping. Thus according to column (1) mode should be 9 or 16. To find out t4g point of maximum concentration the data can be arranged in the shape table as follows:

Analysis Table

Columns | Size of item containing maximum frequency | ||||||

(1) | 9 | 16 | |||||

(2) | 9 | 10 | 15 | 16 | |||

(3) | 8 | 9 | |||||

(4) | 8 | 9 | 10 | ||||

(5) | 9 | 10 | 11 | ||||

(6) | 7 | 8 | 9 | ||||

No. of times a | 1 | 3 | 6 | 3 | 1 | 1 | 2 |

Size occurs |

Since the size 9 occurs the largest number of times it is the model size or mode is 9. If we look at the frequencies in the original table, we shall find that the frequency of 63, which is the maximum single frequency, is against two values, 9 and 16. The series thus appears to be bi-modal but the process of grouping leads us to the conclusion that the concentration of items round 9 is more than the concentration round 16. Even if the frequency against 16 was 64 instead of 63 probably grouping would have disclosed that the concentration of items around about 9 is more, even though the individual frequency against 9 is only 63. It is thus never safe to rely only on the inspection of a series and to locate the mode at the point of maximum frequency. Mode is affected by the frequencies of the neighboring items also, and, therefore, grouping is essential, as it reveals the true point of maximum concentration.

The exact location of the mode of grouped frequency is found by this method (Interpolation formula) Calculating mode of a grouped frequency distribution by Interpolation formula

Class- Interval | 5-10 | 10-15 | 15-20 | 20-25 |

Frequency | 4 | 5 | 7 | 2 |

Step 1:

Here the frequency of class interval 15 – 20 is the maximum.

=> 15-20 = modal class Step 2:

L = Lower limit of modal class =15

fm = Frequency of modal class = 7

fl = Frequency of class preceding the modal class = 5

f2 = Frequency of class succeeding the modal class = 2

h = Size of class interval =10-5 = 5

Step 3:

Mode = L + (fm-fI )/2fm-fl -f2 *h

= 15 + (7-5)/2 *7-5-2 * 5

= 15 + 10/7

= 15 + 1.42 = 16.42

=> Mode = 16.42.

Thus by interpolation formulae, we get 16.42 as the median.

11. Advantages of the Mode

- It is useful in counting the number of times an event occurs.
- It helps in identifying whether an event has occurred more than once.
- It is not affected by extremely large or small values.
- It can be computed in an open-end frequency table.
- It can be located only by inspection in ungrouped data and discrete frequency distribution.
- Mode also is important as it helps us identify the nominal data for various situations. Take the example of the counting of votes. Here, a single-mode value would determine the victory of any particular leader and vice-versa. So, the mode has its own importance and we cannot neglect it at any cost.

Disadvantages

- It is not well defined.
- It is not based on all the values.
- It is stable for large values and it will not be well defined if the data consists of a small number of values.
- It is not capable of further mathematical treatment.
- Sometimes, the data having one or more than one mode, and sometimes the data having no mode at all.

In conclusion, circumstances generally dictate which measure of central tendency—mean, median, or mode—is the most appropriate. If you are interested in a total, the mean tends to be the most meaningful measure of central tendency because it is the total divided by the number of data. For example, the mean income of the individuals in a family tells you how much each family member can spend on life’s necessities. The median measure is good for finding the central value and the mode is used to describe the most typical case.

12. Dispersion

Dispersion measures how the various elements behave with regards to some sort of central tendency, usually the mean. Measures of dispersion include range, inter-quartile range, variance, standard deviation, and absolute deviation. It is a group of analytical tools that describes the spread or variability of a data set.

Definitions

As per A.L. Bowley ‘dispersion is the measure of the variation of items’

As per L.R. Conner ‘dispersion is the measure of the extent to which individual items vary’

13. Importance of the measures of dispersion

- supplements an average or a measure of central tendency
- compares one group of data with another
- indicates how representative the average is.
- reliability of average can be decided with the help of dispersion

14. Methods of dispersion

Some of the methods of dispersion are

- Range- The range is the most obvious measure of dispersion and is the difference between the lowest and highest values in a dataset.
- Inter-quartile range – The inter-quartile range is a measure that indicates the extent to which the central 50% of values within the dataset are dispersed. It is based upon and related to, the median.
- Mean deviation – It is the arithmetic mean of the absolute values of deviation.
- Standard deviation
- Lorenz curve

15. Standard deviation

Since mean can not be further expressed algebraically, a method called standard deviation be used. Standard deviation is a measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance. A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance. Formula for calculating standard deviation For a data set, the mean is

S2 == \(\frac{\Sigma(\mathrm{X}-\mathrm{M}) 2}{\mathrm{n}-1}\)

Where Σ = Sum of

X = Individual score

M = Mean of all scores

N = Sample size (number of scores) Example: To find the Standard deviation of the data set: 3,2,4,1,4,4.

Step 1: Calculate the mean and deviation.

Step 2: Using the deviation, calculate the standard deviation

(0+l+l+4+l+l)/6-l = 8/5

S2 =1.6

S = 1.265

S2 (standard deviation) =1.6

Variance S= 1.265

Short cut Method

The standard deviation that we calculated above formula is correct and will work for calculations, there is another equivalent, shortcut formula that does not require us to first calculate the sample mean. This shortcut formula for the sum of squares is

Σ(x_{i}^{2})-(Σ x_{i})^{2}/n

Here the variable n refers to the number of data points in our sample.

An Example – Standard Formula

We take a sample that is 2, 4, 6, 8. The sample mean is (2 + 4 + 6 + 8)/4 = 20/4 = 5. Now we calculate the difference of each data point with the mean 5.

2 – 5 = -3

4 – 5 = -1

6 – 5 = 1

8 – 5 = 3

We now square each of these numbers and add them together. (-3)2 + (-1)2 + l2 + 32 = 9 + 1 + 1 + 9 = 20.

An Example – Shortcut Formula

Now we will use the same set of data: 2, 4, 6, 8, with the shortcut formula to determine the sum of squares. We first square each data point and add them together: 22 + 42 + 62 + 82 = 4 + 16 + 36 + 64 = 120.

The next step is to add together all of the data and square this sum: (2 + 4 + 6 + 8)2 = 400. We divide this by the number of data points to obtain 400/4 =100.

We now subtract this number from 120. This gives us that the sum of the squared deviations is 20. This was exactly the number that we have already found from the other formula.

The standard deviation plays a dominating role in the study of variation in the data. It is a very widely used measure of dispersion. As far as the important statistical tools are concerned, the first important tool is the mean and the second important tool is the standard deviation.

16. Coefficient of variation

In probability theory and statistics, the coefficient of variation (CV) is a normalized measure of the dispersion of a probability distribution or frequency distribution. It is also known as unitized risk or the variation coefficient. The coefficient of variation (CV) is defined as the ratio of the standard deviation σ to the mean µ:

C_{v} = \(\frac{\sigma}{\mu}\)

The standard deviation of data must always be understood in the context of the mean of the data therefore the coefficient of variation is useful. The coefficient of variation is a dimensionless number. So when comparing between data sets with different units or widely different means, one should use the coefficient of variation for comparison instead of the standard deviation

17. Advantages and disadvantages of standard deviations

Merits

- is based on every item of the data
- takes care of both positive and negative deviation
- is less affected by variation in sampling
- is most popularly used in statistics

Demerits

- is somewhat not easy to understand.
- The extreme values unduly affect the standard deviation

**Measures Of Central Tendency MCQ Questions**

Question 1.

The sum of the mean and the median of 4, 6, 3, 2, 5 is

a. 6.0

b. 9.0

c. 8.0

d. 8.5

Answer:

c. 8.0

Question 2.

The numerical value separating the higher half of a sample from a lower half is

a. Median

b. Mode

c. Mean

d. Standard deviation

Answer:

a. Median

Question 3.

The most frequent value in a data set is

a. Median

b. Mode

c. Arithmetic mean

d. Geometric mean

Answer:

b. Mode

Question 4.

When there is a linear relationship between two numerical variables it can be measured by

a. Mean

b. Mode

c. Scatter diagram

d. Coefficient of correlation

Answer:

d. Coefficient of correlation

Question 5.

Which of the following statements about the median is NOT true?

a. It is more affected by extreme values than the mean

b. It is the middle value that separates the upper half from the lower half

c. It is equal to the mode in bell-shaped distributions.

d. Both b & c

Answer:

a. It is more affected by extreme values than the mean

Question 6.

A measure of central tendency that attempts to describe a set of data by identifying the central position within the data is

a. Mean

b. Mode

c. Median

d. All the above

Answer:

d. All the above

Question 7.

Average is a value that is a representation of a set of data, this definition of average was given by

a. Clark

b. Goxtor

c. Murry R. Spiegal

d. None of the above

Answer:

c. Murry R. Spiegal

Question 8.

Average make it easy for

a. Completing the data

b. For making data short so it becomes manageable

c. Account relations could be formed among variables

d. All of the above

Answer:

d. All of the above

Question 9.

Average can be used

a. Only in unity

b. Can be combined with other average

c. Both a & b

d. None of the above

Answer:

b. Can be combined with another average

Question 10.

The harmonic mean is a part of

a. Positional average

b. Mathematical average

c. Both a & b

d. None of the above

Answer:

b. Mathematical average

Question 11.

From the below mentioned about the mean which is not there in asthmatic mean

a. Assigns equal weightage to both high & small observations used for averages

b. Can not be determined by graphical location

c. It can not use fro data that is not measurable

d. All of the above

Answer:

d. All of the above

Question 12.

Find the median of the below-mentioned observation 15, 20, 45, 30, 60, 36

a. 32

b. 33

c. 47.5

d. None of the above

Answer:

b. 33

Question 13.

The formulae for calculating the median if individual observations are given will be

a. (N+1)/2th if n is add

b. (N+1)th/2 if n is add

c. (N/2)th if n is odd

d. (both a & c

Answer:

a. (N+1)/2th if n is add

Question 14.

For calculation of median through interpolation formulae, “h” in the formulae is

a. Frequency of the median

b. The lower limit of the median class

c. width of the median class

d. Median

Answer:

c. width of the median class

Question 15.

It is not uncommon in a median

a. To locale it graphically

b. To use it then data is qualitative

c. To use it when data is rigidly defined

d. All of the above

Answer:

d. All of the above

Question 16.

The object of statistical averages is to

a. help in comparison

b. help in delusion making

c. helps in the collection of data

d. Both a &b

Answer:

a. help in comparison

Question 17.

In the interpolation formulae used for calculation of mode “h” is

a. Mode

b. width of modal calls

c. Frequency of modal class

d. Lover limit of modal class

Answer:

b. width of modal calls

Question 18.

The mode can be

a. Located graphically

b. Can be computed in an open-and frequent distribution.

c. can be located by just inspection

d. Both B & C

Answer:

d. Both B & C

Question 19.

Only _______ can be used for all algebraic calculations

a. Mean

b. Made

c. Median

d. All of the above

Answer:

d. All of the above

Question 20.

All of the data is highly affected by extreme observation then which type of average it is

a. Mean

b. Mode

c. Median

d. All of the above

Answer:

c. Median

Question 21.

When we want to find the most common data that is being displayed in the series than the best way of calculating it is

a. mean

b. mode

c. media

d. All of the above

Answer:

b. mode

Question 22.

co-efficient of variation is

a. absolute variation

b. static

c. relative variation

d. mane of the above

Answer:

c. relative variation

Question 23.

The following is arranged in ascending order. If the median of the data is 63, find the volume of x in a series 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95,

a. 60

b. 62

c. 63

d. 64

Answer:

b. 62

Question 24.

The number of Wickets taken by a team in series of 10 matches are 2, 3, 4, .5, 0, 1, 3, 3, 4, 3 find the mode of the set

a. 3

b. 2.8

c. 3.3

d. 2.9

Answer:

a. 3

Question 25.

The average (arithmetic mean) of a set of seven numbers is 8. When an eighth number is added to the set, the average of the eight numbers is still 8. What number was added to the set?

a. 6

b. 7

c. 8

d. 9

Answer:

c. 8

Question 26.

If x is the average (arithmetic mean) of 5 consecutive odd integers, what is the median of integers?

a. 0

b. 1

c. x-2

d. x

Answer:

d. x

Question 27.

The sum of 7 consecutive even integers is 224. What integer has the least value in the list?

a. 16

b. 18

c. 26

d. 29

Answer:

c. 26

Question 28.

1, 3, 5, 8, 10, 13. How many different sums can be made by adding any two different numbers from the list above?

a. 6

b. 8

c. 10

d. 12

Answer:

b. 8

Question 29.

In a list of 37 consecutive integers, the median is 70. What is the largest integer in the list?

a. 96

b. 97

c. 98

d. 99

Answer:

c. 98

Question 30.

The mean is a measure of

a. association

b. location

c. relative location

d. variability

Answer:

b. location

Question 31.

The correlation coefficient is a measure of

a. association

b. location

c. relative location

d. variability

Answer:

b. location

Question 32.

Mathematical averages include.

a. Arithmetic means

b. Median

c. Harmonic mean

d. Both a & c

Answer:

d. Both a & c

Question 33.

The most frequently occurring data value in a data set is the

a. median

b. arithmetic mean

c. population parameter d. mode

Answer:

d. mode

Question 34.

A measure of central location which splits the data set into two equal groups is called the

a. mean

b. mode

c. median

d. standard deviation

Answer:

c. median

Question 35.

The coefficient of variation is

a. the same as the variance

b. a measure of central tendency

c. a measure of absolute variability

d. a measure of relative variability

Answer:

d. a measure of relative variability

Questions 36.

The sum of the deviation of the individual data elements from their mean is always

a. equal to zero

b. equal to one

c. negative

d. positive

Answer:

a. equal to zero

Question 37.

An automobile club has four different types of members: A, B, C, and D, depending on the type of membership. For each member, let X indicate his/her membership type.

a. the arithmetic mean of X is meaningful

b. the mode of X is meaningful

c. A histogram is appropriate to display the distribution of X.

d. A box plot is appropriate to display the distribution of X

Answer:

b. the mode of X is meaningful

Question 38.

In general, which of the following statements is FALSE?

a. the sample mean is more sensitive to extreme values than the median.

b. The sample range (i.e. maximum minus minimum) is more sensitive to extreme values than the median.

c. the sample standard deviation is a measure of spread around the sample mean

d. the sample standard deviation is a measure of central tendency around the median

Answer:

d. the sample standard deviation is a measure of central tendency around the median

Question 39.

For ‘n’ observations x<sub>1</sub>, X<sub>2</sub> x<sub>3</sub> ……….. and X<sub>4</sub> of a distribution. From the following what does X denote

x = \(\frac{X_{1}+X_{2}+\ldots-X_{n}}{n}\)

a. Arithmetic mean

b. Median

c. Mode

d. None of the above

Answer:

a. Arithmetic mean

Hint

Arithmetic mean = sum of all observations/Total number of observations

Question 40.

The average friend request for a weak excluding Sunday on Facebook was 10. On Sunday, the average of all 7 days rose to 15. How. many friends visited you on Sunday.

a. 45

b. 15

c. 35

d. 20

Answer:

a. 45

Hint

Mean for 6 days = 10

Mean for 7 days = 15

Total friend request in 6 days = 60

Mean for 7 days = total number of friend requests/total number of days

Mean for 7 days = total number of friend request (total friend request in 6 days + number of friend requests

on Sunday)/total number of days

15= 60 + number of a friend requests on Sunday/7

105 = 60 + number of friend request on Sunday

105-60 = number of a friend requests on Sunday

45= number of a friend requests on Sunday

Question 41.

In a normal distribution-

a. Mode» 3 Median-2 mean

b. Median=3 Mode-2 Mean

c. Mean= 3 Median-2 Mode

d. None of the above

Answer:

a. Mode» 3 Median-2 mean

Hint

Mode = 3 Median – 2 Mean

Question 42.

Coefficient of variation of a distribution is 12.5% and standard deviation is 100. The value of mean will be

a. 100

b. 300

c. 500

d. 800

Answer:

d. 800

Hint

mean = SD/CV × 100 = 100/12.5 × 100 = 800

Question 43.

Two distributions with 100 and 200 items have a mean of 20 and 10 respectively. The combined mean of two distributions will be:

a. 30.50

b. 10.16

c. 13.33

d. 11.12.

Answer:

c. 13.33

Hint

Combined mean = n_{1}x̄_{1} + n_{2}x̄_{2}/n_{1} + n_{2}

= 100 × 20 + 200 × 10/100+200 = 4000/300 = 13.33

Question 44.

A histogram can be used to estimate graphically the value of:

a. Mean

b. Median

c. Mode

d. Upper quartile

Answer:

c. Mode

Hint

The mode is the value that appears most often in a set of data. Mode can be calculated graphically using histogram.

Question 45.

A Lorenz Curve is used to measure:

a. Correlation

b. Variation

c. Arrangements of frequencies

d. Association of attributes.

Answer:

b. Variation

Hint

It is a graphical method.

Question 46.

In a given distribution the value of coefficient of variation is 80%, the value of arithmetic mean is 20. The value of standard deviation would be:

a. 16

b. 20

c. 36

d. 80

Answer:

a. 16

Hint

C.V. * S.D./mean × 100

S.D. = C.V. × mean/100

= 80 × 20/100 =16

Question 47.

The square, root of the arithmetic mean of the squared deviations of items taken from arithmetic mean is called:

a. Mean deviation

b. Quartile deviation

c. Standard deviation

d. Variance

Answer:

c. Standard deviation

Hint

The square, root of the arithmetic mean of the squared deviations of items taken from arithmetic mean is called

For a data set, the standard deviation is S2 = \(\frac{\Sigma(X-M) 2}{n-1}\)

Where Σ = Sum of

X = Individual score

M = Mean of all scores

N = Sample size (number of scores)

Question 48.

In moderately symmetrical distribution, the mode is 49 and median is 44. The value of mean will be:

a. 43

b. 57.3

c. 46

d. 58.0

Answer:

b. 57.3

Hint

Mode = 3 Median – 2 Mean

2Mean = 3 median – Mode

2 mean = 3 × 44 -40 = 132-40 =92

Mean = 92/2 =46

Question 49.

In a moderately a symmetrical distribution. Arithmetic mean = 50, and Mode = 37.5. The value of median will be—

a. 45.83

b. 42.15

c. 43.20

d. 44.00

Answer:

a. 45.83

Hint

Mode = 3 Median – 2 Mean

3Median = mode+2 Mean

= 37.5 +2 × 50

Median =137.5/3 = 45.83

Question 50.

The sum of deviations of a set of observations is zero when the deviations are taken from their –

a. Mode

b. Median

c. Arithmetic mean

d. None of the above

Answer:

c. Arithmetic mean

Hint

The sum of deviations of a set of observations is zero when the deviations are taken from their mean.

Question 51.

The mean and standard deviations of 10 observations are 35 and 2 respectively. If each observation is increased by 4, the changed mean and standard deviation respectively will be-

a. 35 and 2

b. 40 and 4

c. 39 and 2

d. None of the above

Answer:

c. 39 and 2

Hint

If each observation is increased by 4 then mean will be old mean +4 as mean is dependent on change of origin. Thus new mean is 35+4 = 39. But S.D. is not dependent on origin so new S.D. will be same as old S.D.

Question 52.

Mean of first 10 natural numbers is:—

a. 5.5

b. 6.5

c. 8

d. 5

Answer:

a. 5.5

Hint

Mean = sum of observations/No. of observations

Sum of first 10 natural numbers is 1+2+3+4+5+6+7+8+9+10=55

Thus Mean = 55/10 = 5.5

Question 53.

A series was given: – 2,3,6,9, …………………….., what will be the median of the following?

a. 3

b. 6

c. 9

d. 4.5

Answer:

d. 4.5

Hint

Median = n/2th term + (n/2 +1)th term/2 = 4/2 + (4/2+ 1)/2 = 2nd term + 3rd term/2 = 3+6/2 = 4.5

Question 54.

Standard of variance (15, 20, 25), find the mean:-

a. 20

b. 30

c. 40

d. 50

Answer:

a. 20

Hint

Mean = sum of observations/number of observations = 60 (15+20+25)/3 = 20

Question 55.

C.V. is 1600, then find Standard Deviation if A.M is 2.5

a. 40

b. 50

c. 60

d. 70

Answer:

a. 40

Hint

C.V. = S.D./mean × 100

S.D. = C.V. × mean/100

S.D. = 1600 × 2.5/100

= 40

Question 56.

Pie – Chart is:

a. 90°

b. 180°

c. 360°

d. 60°

Answer:

c. 360°

hint

Pie chart is a circle so 360 degree.

Question 57.

The average of 7 numbers is 27. If we include one more number, then average becomes 25. Find the included number.

a. 11

b. 13

c. 15

d. 14

Answer:

a. 11

hint

Average = sum of observations/No. of observations

Sum of 7 numbers = 27 x 7 =189

After adding one more number say x new sum is 189+x .

Average = 189+X/8

25 = 189+X/8

X = 11

Question 58.

When mean, median & mode are equal, then it is a situation of –

a. Positive skewness

b. Negative skewness

c. Symmetrical

d. None of these

Answer:

c. Symmetrical

Hint

According to symmetrical skewness principle A.M. = Median =Mode

Question 59.

The Ogive is formed by-

a. Cumulative frequency

b. Central tendency

c. Both (a) & (b)

d. None of the above

Answer:

a. Cumulative frequency

Hint

Cumulative frequency curve – Cumulative histograms, also known as ogives, are graphs that can be used to determine how many data values lie above or below a particular value in a data set.

Question 60.

When does the value of mean changes-

a. Change of scale

b. Change of origin

c. Both (a) & (b)

d. None of the above

Answer:

c. Both (a) & (b)

Hint

The value of mean changes by change of scale and change of origin.

Questions 61.

Graphical representation of cumulative frequency distribution is known as –

a. Mean

b. Median

c. Mode

d. None of the above

Answer:

b. Median

Hint

Graphical representation of cumulative frequency distribution is known as median.

Question 62.

Find mean of 0.3,5,6,7,9,12,0.2?

a. 5.34

b. 6

c. 5.64

d. 7

Answer:

c. 5.64

Hint

Mean = sum of observations/no. of observations = 39.5/7 = 5.64

Question 63.

Find SD of 10,10,10,10,16,16,16,16.

a. (8) 10

b. 16

c. 3

d. 2

Answer:

c. 3

Mean = sum of observations/no. of observations = 104/8 = 13

Deviation = individual unit – mean = 10-13 =-3

Similarly deviation is -3,-3,-3, -3 ,3 ,3 ,3 ,3

S.D. = \(\sqrt{\frac{\sum \text { deviation }^{2}}{\mathrm{n}}}\)

= \(\sqrt{\frac{9+9+9+9+9+9+9+9}{8}}\)

= 3

Question 64.

Standard deviation is a measure of:

a. Central tendency

b. Symmetry

c. Probability

d. Dispersion

Answer:

d. Dispersion

Hint

Standard deviation is a measure of the dispersion of a set of data from its mean.

Question 65.

Which of the following would be appropriate average for determining the average size of readymade garments:

a. Geometric Mean

b. Mode

c. Arithmetic Mean

d. Median

Answer:

b. Mode

Hint

The mode is the value that appears most often in a set of data. Thus for determining the average of ready made garment this is the best method.

Question 66.

The mean of first 10 even numbers is:

a. 10

b. 9

c. 12

d. 11

Answer:

d. 11

Hint

Mean = sum of observations/no. of observations = 110 (2+4+6+8+10+12+14+16+18+20)/10 = 11

Question 67.

What is the major assumption that one makes, when computing a mean from a grouped data?

a. No value shall occur more than once

b. Each class contains exactly the same number of values

c. All values are discrete

d. Every value in a class interval is equal to its midpoint.

Answer:

c. All values are discrete

Question 68.

If the variance of a sample is 1600, then the value of standard deviation would be:

a. 40

b. 10

c. 1600

d. 160

Answer:

a. 40

Hint

S.D. = \(\sqrt{\text { Variance }}\)

= \(\sqrt{\text { 1600 }}\)

= 40

Question 69.

The mean of first 10 even numbers is:

a. 12

b. 11

c. 10

d. 9

Answer:

b. 11