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Arithmetic mean: The arithmetic mean is the most commonly used measure of central tendency. The mean represents the central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X. 

• In general, if there are N observations as X1, X2, X3, ..., XN, then the Arithmetic Mean is given by

Mean= \({X1 + X2 + X3 +...+ Xn}\over n\).

• This will be written in simpler form without the index i.
• Thus mean = \(ΣX\over N\) where, ΣX = sum of all observations and N = total number of observations. 
• Consider it: The mean of 1, 2, 3, 4, is \(1+2+3+4+5\over 5\) = 3. Let us add '2' to each term, we get, \(3+4+5+6+7\over 5\) = 5
• It should be noted that on adding '2' to each team, the average is also increased by 2.

Hence, we conclude that when a constant figure is added to each individual score, the mean value will be added by the same figure.

Weighted arithmetic mean is used whenever the relative importance of the items in a series differs .

Median is a Positiona average 

Quartiles divides the data into 4 equal parts 

death rate & birth rate (c)Both

Measures of central tendency  are known as averages of first order 

Lorenz Curve  is a geometric method of measuring variability 

Dispersion is the extent to which values in a distribution differ from the average of the distribution. To quantify the extent of the variation, there are certain measures namely:

• Measures of Dispersion From Average:
  • ​The mean deviation (MD) of a series is the arithmetic average of the deviation of various items from a measure of central tendency (mean, median and mode).
  • Standard Deviation (S. D.): One of the most stable measure of variability, it is the most important and commonly used measure of dispersion. It measures the absolute dispersion or variability of a distribution. Standard deviation is the positive square root of the mean of the squared deviations of observations from their mean. It is denoted by S.D. or σ_x.
• Measures Based Upon Spread of Values:
  • Range: It is the simplest measure of variability both from the point of view of calculation and understanding. It is the difference between the largest and the smallest observations in the data. For example, the range of scores 2, 5, 6, 4, 12, 10, 9 and 8 is 12-2 = 10.
  • Interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. It divides a rank-ordered data set into four equal parts. Q1 is the mid-value in the first half of the rank-ordered data, Q2 is the score that has exactly two quarters (median). Q3 is the mid-value in the second half of the rank-ordered data. 
    For example, consider the number 3, 4, 6, 9, 11, 12, 14, 15.
    Q1 is the middle value in the first half of the data set.
    The middle value is the average of the two middle values; that is, Q1 = \(4+6\over 2\) = 5.
    Q3 is the middle value in the second half of the data set.
    i. e. Q3 = \(12+14\over 2\) = 13
    Semi Quartile Range = \(Q3 - Q1\over 2\) = \(13 - 5 \over 2\) = 4

(a) All people

Skewness means lack of symmetry  

Co. Efficient of variation is equal to  (S.D/Mean)x100

In Standard deviation , deviations are taken only from Mean values of series 

Freehand curve is the easiest of all the methods for measuring trend

Square of Standard Deviation is known as Variance

Variance =  (S.D)₂

 (c) Coefficient of variation 

H.M is used in averaging rates , times etc 

 Range & Mean (c)Both

Standard Deviation was First used by Karl Pearson

Variability in the distribution of earth and income is generally measures in terms of Mean Deviation

To find the mode, or modal value, it is best to put the numbers in order. Then count how many of each number. A number that appears most often is the mode.

Concept:

Median: It is the middle value in the given set, while the set is either in increasing or decreasing order.

If the number of observation (n) is odd then median is (n+1)/2^th term.

If the number of observation (n) is even then the median is average of n^th and (n+1)^th term.

Calculation:

Given observation are 5, 6, 11, 23, 4, 8, 20, 2

Increasing order of given data: 2, 4, 5, 6, 8, 11, 20, 23.

The number of observation is even. (n = 8)

So, the median is theaverage of n^th and (n+1)^th term or average of 4th and 5th term

Median \(= \frac{6+8}{2}\)

= 7

Geometric mean is useful in  Both

Harmonic Mean is useful for computing average rate of increase of profits , average rate of speed , average price ..etc  

3 mode of the following series 2 3 4 3 4 3 5 3 7 

When the distribution is of open end classes  Median may appropriate

Mode is not affected by extreme items  

Mode is the value of item of a series which occurs most frequently  

Median divides the value of a variable into two equal parts.

6 for the following observation ; 3,4,6,7,10

Mean is the most commonly used measure of central tendency 

 Assumed mean method 

Geometric mean considered to be the best average in the construction of  Index numbers

Average is a figure that represents the whole group 

There are equal number of observations on the right and on the left of  median

Statistics is an art as well as  Science

The word statistics have been derived from the Status

Statistics does not deals with Qualitative

(d)all of these

Statistics are the eyes of Govt. administration 

(a)precise and accurate 

Concept:

Mode: The mode of the set of data is the value that appears most often.

For example, Data: 2, 5, 5, 6, 7, 4, 3

Mode = 5

Calculation:

We have 2, 4, 5, 4, 5, 6, 2, 4, 3, 6, 3

Clearly, 4 is most occurring in the given observation.

So, mode is 4.

Concept:

Mean: It is the sum of all observations upon a number of observations.

If x1, x2, x3, …., xn are the observations, then mean is given as

\(\rm \bar{X} = \frac{sum\ of\ observations}{number\ of\ observations}\)

\(\rm \bar{X}= \frac{x_{1}+x_{2}+...+x_{n}}{n}\)

Calculation:

We have X̅ = 44 and observations 5, 7, 6, 11, x, 13.

So, mean \(\bar{X} = \frac{5+7+6+11+x+13}{6}\)

⇒ 44 × 6 = 42 + x

⇒ x = 222

When first quartile is 10 , third quartile is 20 , value of quartile deviation is 5  

When the measure of kurtosis is greater than 0 the distribution is leptokurtic

The arithmetic mean of a variable ‘x’ is denoted by the ¯x

 In olden days statistics also called science of kings.

d. All of these

In plural sense, statistics refers to numerical statements of facts.