Which central tendency of distribution divides the distribution into equal parts?

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Page No 103:

Question 1(A):

Fill in the blanks from the alternatives given in the bracket

1) ______ is a positional average.
(Mean/ mode/ median/ geometric)

2) The arithmetic mean of the following observation 4, 8, 12, 16, is ______.
(8, 10, 6, 12)

3) Cumulative frequency is needed while finding the ______ of the distribution.
(mode/ median/ average/ normal)

4) _______ is the item having highest frequency.
(mean/ mode/ median/ average)

5) The second quartile is known as _______.
(median/ lower quartile/ upper quartile/ none)

6) The values which divide the total number of observations into 10 equal parts are ______.
(Quartiles/ percentiles/ deciles/ none)

7) There are ______ deciles.
(7/ 8/ 9/ 10)

8) There are _____ percentiles.
(100/ 98/ 97/ 99)

9) The range of 10 20 30 40 is ______.
(15/ 30/ 10/ 40)

10) 25th percentile is equal to ______.
(Ist quartile / 25 the quartile / 24 the quartile / 2nd quartile)

Answer:

1.

is a positional average.
Explanation
The median is usually the middlemost observation of any series; it is not based on all observations.

2. The arithmetic mean of the following observation 4, 8, 12, 16, is

.
Explanation
Arithmetic mean

X=Σ Xn X=4+8+12+164=404=10

3. Cumulative frequency is needed while finding the medianof the distribution.
Explanation
In case of calculation of median after arranging the data in ascending or descending order, the cumulative frequencies are calculated. We then add the simple frequency as ∑f ad denote as N. Median is then calculated as the value of the item corresponding to the N2+1th item.

4.

is the item having highest frequency.
Explanation
Any observation occurring maximum number of times or most frequently in a series is the mode of that series. Thus, we can say that the mode is the item having the highest frequency.

5. Second quartile is known as

.
Explanation
The second quartile divides a series into two equal parts. Median, being the middlemost observation of any series, divides the series into two equal parts. Thus, the second quartile is also known as the median.

6. The values which divide the total number of observations into 10 equal parts are

.
Explanation
Deciles are the values that divide the data into 10 equal parts. For this, the data are first arranged in ascending order. Deciles then divide the data into 10 groups of equal size.

7. There are 9 deciles
Explanation
Since deciles divide the data into 10 equal groups. Therefore, there are 9 points in the entire series which break the data into 10 equal groups.

8. There are 99 percentiles.
Since percentiles divide the data into 100 equal groups. Therefore, there are 99 points in the entire series which breaks the data into 100 groups of equal size.

9. The range of 10 20 30 40 is 30.
Explanation
Range = Highest value – Lowest value
Highest value = 40
Lowest value = 10
Range = 40 – 10 = 30

10. 25th percentile is equal to

.
Explanation
A quartile divides the data into 4 equal groups. Therefore, the 1st quartile would mean 1/4th of the entire data. Similarly, a percentile divides the data into 100 equal groups. Now, 25th percentile would also mean 1/4th of the entire data. Thus, 25th percentile is equal to the 1st quartile.

Page No 103:

Question 1(B):

Match the following:
 

Answer:

i. The mode is the observation occurring maximum number of times or most frequently in a series.
ii. The median divides the data into two equal parts. Thus, it is the middlemost item of a series.
iii. The mean is the simplest way of representing the entire data. It is calculated by dividing the sum of the given items by the total number of items. Thus, it is precise in nature.
iv. A central tendency is a single average value that lies within the series or data and, at the same time, represents the entire data. It is said to be a group component.
v. The range implies the span of data. In other words, it defines the difference between the largest and smallest values of the given data.

Page No 103:

Question 1(C):

State whether, the following statements are true or false and rewrite them.

1) Mode is the value having maximum frequency.

2) Median is not affected by extreme items.

3) Arithmetic mean is not based on all observations.

4) Mode can be calculated by mere inspection.

5) Mean is not rigidly defined.

6) First decile is greater than first quartile.

Answer:

1. The given statement is true.
Explanation
Any observation that repeats itself the maximum number of times is the mode of that data. Thus, it would be correct to say that the mode is that observation or value that has the maximum frequency.

2. The statement is true.
Explanation
The median is the middlemost value of any set of data arranged in an array. It is just a positional average, thereby, is not affected by the presence extreme values.

3. The given statement is false.
Explanation
The arithmetic mean is the sum total of all observations divided by the total number of observations. It includes all the observations given in the series.

4. The given statement is absolutely true.
Explanation
The mode is the observation or value that repeats itself the maximum number of times or that has the maximum frequency. Thus, it can be easily inspected or located by merely looking at the given data.

5. The given statement is false.
Explanation
The mean is said to be rigidly defined because even if we use any method of calculation of mean, we will get the same result. In other words, the resultant value remains the same, irrespective of the method we use.

6. The given statement is false.
Explanation
Deciles divide data into 10 equal parts, whereas quartiles divide data into 4 equal parts. Therefore, the 1st decile would mean 1/10th of the data, whereas the 1st quartile would imply 1/4th of the data. Hence, we can say that the 1st decile is smaller than the 1st quartile.

Page No 103:

Question 2(A):

Define/Explain the following concepts.

1. Arithmetic mean
2. Median.
3. Mode
4. Quartiles
5. Deciles.
6. Percentiles
7. Range

Answer:

1. Arithmetic mean - The arithmetic mean refers to the value calculated by dividing the sum of the values of the given items by the total number of items present in a series. Mathematically, it is measured as:

X=x1+x2+x3+......+xnN

Here, X is the arithmetic mean.

2. Median - The median is the middlemost value of a set of data when arranged in an array (ascending or descending). The median divides the data into two equal groups. Mathematically, for an individual series, it is measured as:

M = Size of the N+12 th item
Here,
M = Median
N = Total number of items

3. Mode - The mode is the observation or value that repeats itself the maximum number of times or that has the maximum frequency. It is easily measured by merely looking and locating the value with the highest frequency.

4. Quartiles - Quartiles are the values of the given set of data that divide the whole data or the set of observations into four equal segments.
There are three quartiles in data known as the first, second and third quartiles.
Mathematically,

Q1=N+14th observationQ2=2N+ 14th observationQ3=3N+14th observation

Here, N is the total number of items.

5. Deciles - Deciles are the values of the given set of data that divide the whole data into 10 equal groups. There are 9 deciles in a set of data that divide the set of observations into 10 parts of the same size, namely D1D2D3D4D5D6D7D 8D9.
Mathematically,
D1=N+110th item

D2=2 N+110th item

D3=3N+110th item

D4=4N+110th item

D5=5N+110 th item

D6=6N+110th item

D7= 7N+110th item

D8=8N+110th ite m

D9=9N+110th item

Here, N is the total number of items.

6. Percentile - A percentile divides the given set of data into 100 equal parts. There are a total of 99 percentiles that help in dividing the data into 100 parts by giving 99 dividing points, namely P1P2P3.............P99.
Mathematically,
P 1=N+1100th item

P2=2N+1100th it em

P3=3N+1100th item
.
.
.
.
P99=99N +1100th item
Here, N is the total number of items.

7. Range - The range is defined as the span of distribution. It is calculated by finding the difference between the smallest and largest values of the given set of distribution.
Range = Largest value – Smallest value
Or,
R = L – S

Page No 103:

Question 2(B):

Give reasons.

1) Arithmetic mean is measure of central tendency.

2) Mode is that value which has maximum frequency.

3) Mode has a number of merits.

4) Median divides the series into two equal parts.

5) Mean has not any limitations.

6) Median is not affected by remote values.

7) Percentiles divide the data into hundred equal parts.

Answer:

1. The central value represents the entire data in the sense that the values of observations in the data lie close to the central value. Arithmetic mean is the average of all items in the series. It is based on all the items in the data. Thus, it can be interpreted as a value that is an indicative of the various items in the data. Hence, we can say that arithmetic mean is a measure of the central tendency.

2. The mode is the observation or value that repeats itself the maximum number of times in the given series. Here, the frequency represents the number of times the value is repeated. Thus, it is correct to say that the mode is the observation or value that has the maximum frequency.

Example: The following marks are scored by 10 students in a class. Find the modal marks.

A close examination of the data reveals that 15 occurs the highest number of times in the series i.e. 4 times. Thus, 15 is the modal marks.

3. Any observation that repeats itself the maximum number of times is called the mode of that data. The following are some of the merits of mode:
i. It is one of the simplest measures of central tendency and can be calculated by the mere inspection of the series.
ii. It can be presented graphically.
iii. It is not affected by the extreme values of the series.
iv. Calculation of mode does not require all the details about the series. It can be calculated for open-ended classes as well.

4. The median is the middlemost value of a set of data when it is arranged in an array (ascending or descending). Half of the items lie after the median and half of the items lie before the median; thus, the median divides the entire series into two equal parts.

5. The mean has both merits and demerits. The following are a few limitations of the mean:
i. It is largely affected by the extreme values of the series.
ii. Because it is based on all observations of the series, it cannot be calculated for open-ended classes.
iii. Sometimes, it gives absurd results, which are impractical.

6. The median is the middlemost value of a set of data when it is arranged in an array (ascending or descending). It is just a positional average that is based on the number of observations in the series and not on the values of those observations. In other words, it is the number of observations and not the values of the observations that affect the median. Thus, we can say that the median does not get affected by the remote values.

7. A percentile divides the given set of data into 100 equal parts. There are a total of 99 percentiles that divide the data into 100 parts, giving 99 dividing points. Each group contains an equal number of observations, thus giving 100 groups of equal size. Thus, we can say that percentiles break data into 100 equal parts.

Page No 103:

Question 3(A):

Distinguish between

1) Arithmetic mean and Mode
2) Arithmetic mean and median
3) Median and mode
4) Quartiles and Deciles
5) Deciles and Percentiles
6) Quartiles and Percentiles

Answer:

1.

2.

3.

4.

5.

6.

Page No 103:

Question 3(B):

Write short notes

1) Merits and demerits of mode.

2) Write merits and demerits of median.

3)  Define quartile- Give the formula for computing (1) Q1, (2) Q2, (3) Q3

4) Define deciles. Give the formulas for computing (1) D5, (ii) D7 (iii) D9

5) Define percentiles. Give the formulas. For computing the 3rd  percentiles and 99th percentiles.

6) Give definitions

(1) Median (2) Mode

Answer:

1. The following are the merits and demerits of the mode:

Merits:

i. It is easy to calculate.
ii. It is not influenced by extreme values.
iii. It is useful in the analysis of comprehensive data.
iv. It can be graphically determined.
v. Its value can be measured in open-ended class intervals as well.
vi. It can be determined or measured by inspection or location only.

Demerits:

i. The value of mode is ambiguous and indefinite in the sense that a series can even have two modes.
ii. It is not based on all observations.
iii. It is a mere positional average rather than arithmetic average.
iv. Its scope is limited in small samples.
v. It is an unstable average.

2. The following are the merits and demerits of the median:

Merits:

i. It is easy and simple to calculate.
ii. It is not influenced by extreme values.
iii. All details of data are not required in the calculation of median.
iv. It can be determined by mere inspection and graphically as well.
v. It is considered an ideal measure for studying qualitative attributes.

Demerits:

i. It is not based on all observations.
ii. It is not suitable for further algebraic treatment.
iii. It is not a true representative of data if the data lack uniformity.
iv. It is greatly affected by sampling fluctuations.
v. It ignores the extreme values of data.

3. Quartiles are the values of a given set of data that divide the whole data or set of observations into four equal parts. The following are the formulas for computing the three quartiles:

Q1= N+14th observation

Q2=2N+14th  observation

Q3=3N+14th observation

4. Deciles are the values of a given set of data that divide the whole data or set of observations into 10 equal parts. There are 9 deciles in data that divide it into 10 parts of the same size, namely  D1D2D3D4D5D6D7 D8D9.
The following are the formulas for D5, D7 and D9:

D5=5N+110th item

D7=7N+110th item

D9=9N+19th  item

5. Percentiles are the values of a given set of data that divide the whole set of observations into 100 equal parts. There are 99 percentiles that give 99 dividing points denoted by P1P2P3........... P99.
The following are the formulas for computing the 3rd and 99th percentiles:

P3=3N+1100th itemP99= 99N+1100th item

6. (1) The median is the middlemost value of a given set of data arranged in an array (ascending or descending). It divides the data into two equal groups. Mathematically, for an individual series, it is calculated as:
M = Size of the N+12th item
Here,
M = Median
N = Total number of items

(2) The mode is the observation or value that repeats itself the maximum number of times or that has the maximum frequency. It is easily measured by merely looking and locating the value with the highest frequency.

Page No 103:

Question 4.1:

Write answers for the following questions

Explain the concept of central tendency.

Answer:

Central tendency refers to a central value or a representative value of a statistical series. The central value represents the entire data in the sense that the values of observations in the data lie close to the central value. Thus, central value can be said to represent the entire data set. A central value makes the raw data easy to understand and analyse, thereby draw clearer conclusions.

Page No 103:

Question 4.2:

Write answers for the following questions

Define arithmetic mean-state the merits and demerits of the mean.

Answer:

The arithmetic mean refers to the value calculated by dividing the sum of the values of the given items by the total number of items in a series. Mathematically,

X=x1+x2+x3+......+xnN
Merits:

i. Simple to calculate: The arithmetic mean is simple and easy to calculate, as it is measured using a simple formula.
ii. Rigidly defined: It implies that the resultant value remains the same, irrespective of the method used for calculation.
iii. Sample fluctuations: The mean is stable in nature; it is not much affected by fluctuations in sampling.

Demerits:

i. Absurd results: It is often seen that the mean gives absurd or unreliable results, which are unrealistic in nature.
ii. Inaccurate in some cases: The mean is not suitable in case the data given is incomplete. This is because if even a single item or value is missing, the mean loses its accuracy.
iii. Affected by extreme values: The mean gets affected by extreme values, as this measure is based on all the items of the given data.

Page No 103:

Question 4.3:

Write answers for the following questions

Write the drawbacks of the median.

Answer:

The following are some drawbacks of the median:

i. Not based on all observations: The median being the middlemost value is not based on all observations. It is said to be a positional average and hence is not dependent on all values.
ii. Essential array: To calculate the median, it is essential to form an array first. However, if a series is lengthy, then the array is difficult to form.
iii. Ignores extreme values: The median ignores extreme values; thus, it is less useful in cases where large weights are assigned to extreme values.
iv. Faulty representative: Because the median ignores extreme values and is not based on all observations, it is not a true representative of the series.

Page No 103:

Question 4.4:

Write answers for the following questions

Write merits and demerits of range.

Answer:

The range is defined as the difference between the smallest and largest values of a given set of distribution. The following are the merits and demerits of the range:

Merits
a) Easy to understand: It is a very simple and easy-to-understand method of finding averages. It basically depicts the span of the given series.
b) Faster to calculate: With a simple formula, range is easily calculable. It is just measured by taking the difference between highest and lowest value of the given series.

Demerits
a) Depends on two values: For calculating a range, only two values are required- highest and lowest. Thus, it does not take into consideration other values given in the series.
b) Affected by sampling: It is considerably affected by fluctuations in samples and distributions.
c) Indeterminate in open-ended class intervals: It cannot be determined in open-ended class intervals, as it is difficult to find out the highest and lowest values of a series.

Page No 103:

Question 5.1:

Do you agree or disagree with the following statements? Give Reasons.

Median is the value of the most frequent item in the series.

Answer:

Disagree

We do not agree with the statement. The median is the middlemost value of a given set of data arranged in an array (ascending or descending). It is the mode that depicts the most frequent item in a series.

Page No 104:

Question 5.2:

Do you agree or disagree with the following statements? Give Reasons.

Mean is the middle item of the series.

Answer:

Agree

The statement is wrong. The arithmetic mean is the value calculated by dividing the sum of the values of the given items in a series by the total number of items in the series. The median, on the other hand, depicts the middle item of the given series.

Page No 104:

Question 5.3:

Do you agree or disagree with the following statements? Give Reasons.

Mean is a measure of central tendency.

Answer:

Agree

We agree with the given statement. The mean is the value calculated by dividing the sum of the values of the given items in a series by the total number of items in the series. It is one of the measures of central tendency and is a representative of the whole set of data.

Page No 104:

Question 5.4:

Do you agree or disagree with the following statements? Give Reasons.

Mode is not the value of an item which occurs most frequently in a series.

Answer:

Disagree

No, we do not agree with this statement. The mode is the value of an item that has the maximum frequency in a series or that occurs most frequently in a series.

Page No 104:

Question 5.5:

Mean and mode means the same thing.

Answer:

Disagree

No, we disagree with the statement. The mean is the value obtained by dividing the sum of the values of the given items in a series by the total number of items in the series. Mode, on the other hand, refers to the value that repeats itself the maximum number of times or that has the maximum frequency.

Page No 104:

Question 5.6:

Do you agree or disagree with the following statements? Give Reasons.

There is no difference between quartile, decile and percentile.

Answer:

Disagree

No, we completely disagree with the statement. This is because the three concepts are very different from each other. Quartiles are the values of a given set of data that divide the whole data or observations into 4 equal parts. Deciles divide the given set of data into 10 equal groups. Percentiles are the values that divide the whole set of data into 100 equal parts. The numbers of quartiles, deciles and percentiles are 3, 9 and 99, respectively.

Page No 104:

Question 6.1:

Answer the following questions in detail−

The following are the monthly income of 10 families in a village.

Calculate mean average income per family.
Rs 550, Rs 650, Rs 700, Rs 400, Rs 550,
Rs 650, Rs 800, Rs 900, Rs 450, Rs 350

Answer:

Mean=X=ΣxnX=550+650+700+400+550 +650+800+900+450+35010=600010=600

Thus, mean income is Rs 600.

Page No 104:

Question 6.2:

Answer the following questions in detail−

Calculate the arithmetic mean from the following

Answer:

X=∑xifi∑fi=35450=7.08

Page No 104:

Question 6.3:

Answer the following questions in detail−

In the following data the daily wages is given (in rupees) of 25 workers in a sugar factory. Calculate the mean wages

Answer:

X=∑xifi∑fi=178025=71.2
Thus, the mean wage is Rs 71.2

Page No 104:

Question 6.4:

Answer the following questions in detail−

Calculate the arithmetic mean from the following data using direct method

Answer:

X=∑xifi∑fi=288090=320

Page No 104:

Question 6.5:

Answer the following questions in detail−

Calculate mean from the following data by step deviation method.

Answer:

X=A+∑fdiN×i   =35+120×10   =35.5

Page No 104:

Question 6.6:

Answer the following questions in detail−

Calculate Arithmetic mean From the following data Use short-cut method

Answer:

X=A+∑fdN  =35-110150  = 34.26

Page No 104:

Question 6.7:

Answer the following questions in detail−

Find median from the following data

Answer:

35, 35, 36, 37, 38, 39, 40

          N = 7

Median=N+12th term            =82=4th term           =37

Thus, the median is the 4th item, i.e., 37.

Page No 104:

Question 6.8:

Answer the following questions in detail−

Find median marks obtained by the students out of 50 marks in economics.

Answer:

Median=N2th term           = 482=24th item

As 24 falls in the cumulative frequency of 31, the median is the value corresponding to the 31st C.F.
Hence, the median is 20.

Page No 104:

Question 6.9:

Answer the following questions in detail−

Calculate median from the following data-

Answer:

Median can be calculated using the following formula:

N2th item=402=20th item

20th item falls in the 27th C.F

Median=L1+L2-L1f1×N2-C.F

L1= 30
L2= 40
fi = 10
C.F = 17

=30+40-3010×402-17=30+3=33

Page No 104:

Question 6.10:

Answer the following questions in detail−

Find the mode from the following data.

Answer:

The mode is the value having the highest frequency. Here, the mode is 10, as it has the highest frequency of 4.

Page No 104:

Question 6.11:

Answer the following questions in detail−

Calculate the mode from the following data

Answer:

The mode is the value having the highest frequency. Here, the mode is 73, as it has the highest frequency of 160.

Page No 104:

Question 6.12:

Answer the following questions in detail−

Calculate the mode from the following fequency distribution

Answer:

In the given series, (8–12) is the modal class, as it has the highest frequency of 12.
In such case, the mode is calculated as follows:

Mode=L1+L2-L1 f1- f02 f1-f0-f2        =8+12-8  12-62×12-6-7        =8+2411=5.82

Page No 104:

Question 6.13:

Answer the following questions in detail−

Calculate range from the following data

Answer:

100, 150, 200, 300, 550, 600
Range = Highest value – Lowest value
H = 600 and L = 100
Now,
Range = 600 – 100 = 500

Page No 104:

Question 6.14:

Answer the following questions in detail−

Wage distribution of 100 workers is given below. Calculate range.

Answer:

Range=Highest value - Lowest value          =50-10           =40

Page No 104:

Question 6.15:

Answer the following questions in detail−

From the following data of pocket money given to the students, calculate Q1, Q2 and Q3

50, 40, 60, 90, 80, 70, 100, 140, 130, 120, 110

Answer:

First the given data is arranged in ascending order as follows:

40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140

The three quartiles can be calculated in the following manner:

N=11Q1=N+ 14th observation    =11+14th 3rd observationQ1=60
Q2=2N+14th observation    =2×11+14      =6th observation    =90
Q3=3×N+1 4th observation    =3×11+14th    =9th observation     =120

Page No 104:

Question 6.16:

Answer the following questions in detail−

Calculate Q1 and Q3 from the following data.

Answer:

Q1=N4th=1504th=57.5th

It falls in the 50th c.f

Now,

Q1=30-40Q1=L1+L2-L1fN4-cfQ1 =30+40-30251504-25

Thus, we have:

Q1=35Q3=3N4 th    =3×1504th=112.5

This falls in the 125th c.f

  Q3=50-60Q3=L1+L2-L1f3N4-cfQ3=50+60 -50353×1504-90

Thus, we have:
Q3 = 56.42

Page No 104:

Question 6.17:

Answer the following questions in detail−

From the following frequency distribution calculate first and third quartile.

Answer:

The quartiles can be found using the following formulas:

Q1=N+14th item    =59+ 14=15th

It falls in the 23rd c.f which corresponds to item having value 6.

Thus, Q1 = 6

Q3=3N+14th item=3 ×59+14=45th item.

It falls in the 45th c.f which corresponds to item having value 10.

Thus, Q3 = 10

Page No 105:

Question 6.18:

Answer the following questions in detail−

From the following data calculate D1, D4 and D7

11, 15, 12, 16, 13, 17, 14, 18, 20, 19, 25.
21, 24, 22, 23, 29, 28, 25, 26.

Answer:

The given data is first arranged in ascending order as follows:
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 28, 29
Now, we find the deciles in the following manner:

 N=19D1=N+110th item    =19+110=2nd  item

Thus, D1=12          D4=4N+110th item              =8th

Thus,  D4=18         D7=7N+110th item              =14th itemThus, D7=24

Page No 105:

Question 6.19:

Answer the following questions in detail−

From the following data find out D2, D7 and D9

Answer:

D2=2N10th=16010=16th item

16th item falls in 27th c.f

D2=20+30-201416010-13=22.14D7=7N 10th=56010th=56th item

56th item falls in 70th C.F.

D7=40+50-4018× 56010-52    =42.22D9=9N10th=720 10th =72nd item

72nd item falls in 76th C.F.

D9=50+60-506×72010-70     =53.33

Page No 105:

Question 6.20:

Answer the following questions in detail−

Find D3 and D4 from the following data.

Answer:

D3=3N+110th item    =345+110 =13.8th item

13.8th item falls in 5th C.F which corresponds to X equal to 1

Thus, D3 = 1

D4=4+N+110 th item    =445+110th=18.4th item

18.4th item falls in 20th C.F. which corresponds to X equal to 4

Thus, D4 = 4

Page No 105:

Question 6.21:

Answer the following questions in detail−

Calculate P40 and P70 for the following data.

Answer:

P40=40N100th item     =40×25100th  item     =10th items

10th item falls in 10th C.F.

P40=L1+L2 -L1f40N100-c.fP40=L1+L2-L14 40N100-c.f
P40=20+30-2041000100-6 P40=30P70=70N100th item     =70×25100 th item     =17.5th item

17.5th item falls in 21st C.F.

P70=40+50 -4061750100-215     =44.16

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Which of the measure of central tendency does divide the distribution in two equal parts?

Which measure of central tendency divides a frequency distribution into four equal parts?

Which divides data into two equal parts?

Which measure of central tendency will be equal for a symmetrical distribution?