__MEAN__
The arithmetic mean (or
simply "mean") of a sample is the sum of the sampled
values divided by the number of items in the sample.

__MERITS OF ARITHEMETIC MEAN__

l ARITHEMETIC MEAN RIGIDLY DEFINED BY ALGEBRIC FORMULA

l It is easy to calculate and simple to understand

l IT BASED ON ALL OBSERVATIONS AND IT CAN BE REGARDED AS REPRESENTATIVE OF THE GIVEN DATA

l It is capable of being treated mathematically and hence it is widely used in statistical analysis.

l Arithmetic mean can be computed even if the detailed distribution is not known but some of the observation and number of the observation are known.

l It is least affected by the fluctuation of sampling

__DEMERITS OF ARITHMETIC MEAN__

l It can neither be determined by inspection or by graphical location

l Arithmetic mean cannot be computed for qualitative data like data on intelligence honesty and smoking habit etc

l It is too much affected by extreme observations and hence it is not adequately represent data consisting of some extreme point

l Arithmetic mean cannot be computed when class intervals have open ends

__Median:__The median is that value of the series which divides the group into two equal parts, one part comprising all values greater than the median value and the other part comprising all the values smaller than the median value.

__Merits of median__(1)

**It is very simple measure of the central tendency of the series. I the case of simple statistical series, just a glance at the data is enough to locate the median value.**

__Simplicity:-__(2)

__Free from the effect of extreme values:__**-**Unlike arithmetic mean, median value is not destroyed by the extreme values of the series.

(3)

**- Certainty is another merits is the median. Median values are always a certain specific value in the series.**

__Certainty:__(4)

**- Median value is real value and is a better representative value of the series compared to arithmetic mean average, the value of which may not exist in the series at all.**

__Real value:__(5)

**Graphic presentation:**__- Besides algebraic approach, the median value can be estimated also through the graphic presentation of data.__

(6)

**Possible even when data is incomplete:**__- Median can be estimated even in the case of certain incomplete series. It is enough if one knows the number of items and the middle item of the series.__

**Demerits of median:**

Following are the various demerits of median:

(1)

**- Median fails to be a representative measure in case of such series the different values of which are wide apart from each other. Also, median is of limited representative character as it is not based on all the items in the series.**

__Lack of representative character:__(2)

**Unrealistic:-**__When the median is located somewhere between the two middle values, it remains only an approximate measure, not a precise value.__

(3)

__Lack of algebraic treatment:__**-**Arithmetic mean is capable of further algebraic treatment, but median is not. For example, multiplying the median with the number of items in the series will not give us the sum total of the values of the series.

However, median is quite a simple method finding an average of a series. It is quite a commonly used measure in the case of such series which are related to qualitative observation as and health of the student.

**Mode:**

The value of the variable which occurs most frequently in a distribution is called the mode.

**Merits of mode:**

Following are the various merits of mode:

(1)

__Simple and popular:__**-**Mode is very simple measure of central tendency. Sometimes, just at the series is enough to locate the model value. Because of its simplicity, it s a very popular measure of the central tendency.

(2)

**- Compared top mean, mode is less affected by marginal values in the series. Mode is determined only by the value with highest frequencies.**

__Less effect of marginal values:__(3)

**Mode can be located graphically, with the help of histogram.**

__Graphic presentation:-__(4)

**- Mode is that value which occurs most frequently in the series. Accordingly, mode is the best representative value of the series.**

__Best representative:__(5)

**No need of knowing all the items or frequencies:**__- The calculation of mode does not require knowledge of all the items and frequencies of a distribution. In simple series, it is enough if one knows the items with highest frequencies in the distribution.__

**Demerits of mode:**

Following are the various demerits of mode:

(1)

**- Mode is an uncertain and vague measure of the central tendency.**

__Uncertain and vague:__(2)

**- Unlike mean, mode is not capable of further algebraic treatment.**

__Not capable of algebraic treatment:__(3)

__Difficult:__**-**With frequencies of all items are identical, it is difficult to identify the modal value.

(4)

**- Calculation of mode involves cumbersome procedure of grouping the data. If the extent of grouping changes there will be a change in the model value.**

__Complex procedure of grouping__:(5)

**It ignores extreme marginal frequencies. To that extent model value is not a representative value of all the items in a series. Besides, one can question the representative character of the model value as its calculation does not involve all items of the series.**

__Ignores extreme marginal frequencies__:-##
**TO FIND MEAN**

**TO FIND MEAN**

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2. Following table gives age distribution of people sufering from 'Asthma due to air pollution in certain city. Find mean age of persons suffering from 'Asthma' by 'Direct Method'.

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3. The measurements (in mm) of the diameters of the head of screws are given below: Calculate mean diameter of head of a screw of 'Assumed Mean Method'.

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4. Below is given frequency distribution of marks (out of 100) obtained by the students. Calculate mean marks scored by a student by 'Assumed Mean Method'.

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5. Following table gives frequency distribution of milk (in litres) given per week by 50 cows. Find average (mean) amount of milk given by a cow by 'Shift of Origin Method.'

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6. Following table given frequency distribution of trees planted by different housing societies in a particular locality. Find the number of trees planted by housing society by using 'step deviation method'.

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7. Following table gives age distribution of people suffering from 'Asthma due to air pollution in certain city. Find mean by 'Step Deviation method'.

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8. The measurements (in mm) of the diameters of the head of screws are given below: Find mean by 'Step deviation method'.

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9. Solve by 'Assumed Mean method. Following table gives frequency distribution of trees planted by different housing societies in a particular locality;. Find Mean.

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10. Solve by 'Step Deviation Method. Below is the frequency distribution of marks (out of 100) obtained by the students. Find mean.

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**TO FIND MEDIAN **

### 1. Following is the distribution of the size of certain farms from a taluka (tehasil):

**2. Below is given distribution of profit in Rs. per day of a shop in certain town: Calculate median profit of a shop.**

3. Following table shows distribution of monthly expenditure (in Rs.) done by households in a certain village on electricity: Find median expenditure done by a household on electricity per month.

3. Following table shows distribution of monthly expenditure (in Rs.) done by households in a certain village on electricity: Find median expenditure done by a household on electricity per month.

4. The following table shows ages of 300 patients getting medical treatment in a hospital on a particular day. Find median age of a patient.

4. The following table shows ages of 300 patients getting medical treatment in a hospital on a particular day. Find median age of a patient.

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**TO FIND MODE**

**1. The weight of coffee (in gms) in 70 packets is given below: Determine the modal weight of coffee in a packet.**

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2. Forty persons were examined for their Hemoglobin % in blood (in mg per 100 ml) and the results were grouped as below: Determine modal value of Hemoglobin % in blood of a person.

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3. The maximum bowling speed (Kms/hour) of 33 players at a cricket coaching centre is given below.

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**RELATIONSHIP BETWEEN MEAN, MEDIAN AND MODE**

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1. For a certain frequency distribution the values of Mean and Mode are 54.6 and 54 respectively. Find the value of median.

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2. For a certain frequency distribution the values of Median and Mode is 95.75 and 95.5 respectively, find the mean.