Statics: Averages and Arithmetic Mean, Mode
Types of Averages:
One of the powerful tools of analysis is to calculate or typical value for a given set of data. Such a value a single average value that represents the entire mass is neither the smallest nor the largest value. but it is of data. The word average is very commonly used in a number whose value is somewhere in the middle of day-to-day conversation. An 'average' is a single the group. For this reason an average is frequently value which is considered as the most representative referred as a measure of central tendency.
Objectives of Averaging:
There are two main objectives of finding the averages
- To get one single value that describes the characteristic of the entire data.
- To facilitate comparison.
Characteristics of a Good Average:
Since an average is a single value representing a group of values, it is desirable that such a value satisfies the following properties.
- It should be easy to understand: Since statistics methods are designed to simplify complexity, it is desirable than an averages be such that it can be readily understood; otherwise, its use is bound to be very limited.
- Its should be simple to compute: Not only, an average should be easy to understand but also it should be simple to compute so that it can be used widely. Though ease of computation is desirable, it should not be sought at the expense of other advantages, ie., if in the interest of greater accuracy, use of a more difficult average is desirable, one should prefer that.
- Should be based on all the observations: The averages should depend upon each and every observation so that if any of the observations is dropped average itself is altered.
- It should be rigidly defined: An average should be properly defined so that it has one and only one interpretation. It should preferably be defined by an algebraic formula so that if different people compute the average from the same figures they all get the same answer.
- It should have sampling stability: We should prefer to get a value which has what the statisticians call "Sampling Stability". This means that if we pick 10 different groups of college students, and compute the average age of each group, we should expect to get approximately the same values. It does not mean, however, that there can be no difference in the value of different samples. There may be some difference but those averages in which this difference, technically called sampling fluctuation, is less are considered better than those in which this difference is more.
The following are the important measures of central tendency which are generally used:
- Arithmetic Mean
- Median
- Mode
- Geometric Mean
- Harmonic Mean
(1) Arithmetic Mean:
The most popular and widely used measure for representing the entire data by one value is what most laymen call an "average" and what the statisticians call the arithmetic mean. Its value is obtained by adding together all the observations and then by dividing this total by the number of observations.
Calculation of Arithmetic Mean- Ungrouped Data:
For ungrouped data, arithmetic mean may be computed by applying any of the following methods:
(I) Direct Method:
The arithmetic mean, often simply referred to as mean, is the total of the values of a set of observations divided by their total number of observations. Thus, if X1,X2, .......Xn represents the values of N items or observations, the arithmetic mean denoted by X is defined as
Bar X = X1+X2+......................+Xn/n
Bar X= ∑X/N
Example 1:-
The monthly income (in rupees) of 10 persons working in a firm is as follows:
1487, 1493, 1502, 1446, 1475, 1492, 1572, 1516, 1468, 1489. Find the average monthly income.
Solution:
Let income be denoted by x.
∑ x= 1487+1493+1502+1446+1475+1492+1572+1516+1468+1489= 14,940
Bar x= ∑X/n = 14,940/10 = 1494.
(2) Median:
The median is the measure of Central Tendency which appears in the "middle" of an ordered sequence of values. That is, half of the observations in a set of data are smaller than it and half of the observations are greater than it.
As distinct from the arithmetic mean which is calculated from the value of every observation in the series, the median is what is called a 'positional' average. The term 'Position' refers to the place of value in a series. The place of the median in a series is such that an equal number of observations lie on either side of it. Median is thus the central value of the distribution of the value that divides the distribution into two equal parts. If there are even number of observations in a series, there is no actual value exactly in the middle of the series and as such the median is indeterminate. In his case the median is arbitrarily taken to be half-way between the two middle observations.
(3) Mode:
Mode is the most typical or commonly observed value in a set of data. For example, if we take the values of six different observations as 5, 8, 10, 8, 5, 8 mode will be 8, as it has occurred maximum number of times, i.e., 3 times. Graphically, it is the value on the X-axis below the peak, or highest point, of the frequency curve as can be seen from the following.
There are many situations in which arithmetic mean and median fail to reveal the true characteristic of data. For example, when we talk of most common wage, most common income, most common height, we have in mind mode and not the arithmetic mean or median discussed earlier. The mean does not always provide an accurate reflection of the data due to the presence of extreme values. Median may also prove to be quite unrepresentative of the data owing to an uneven distribution of the series. Both these shortcoming may be overcome by the use of mode. Mode refers to that value which occurs most frequently in a distribution. Mode is the easiest to compute since it is the value corresponding to the highest frequency.
(4) Geometric Mean:
Geometric Mean is defined as the N root of the product of N observations of a series. If there are two observations, we take the square root, if there are three observations, the cube root; and so on.
Symbolically, ∑ log x = 10.1259
Where X1, X2, ..........Xn refer to the various observations of the series.
(5) Harmonic Mean:
The Harmonic Mean, like the Arithmetic and Geometric mean, is computed from all observations. It is useful in special cases for averaging rates. However, Harmonic Mean cannot be computed when there are both positive and negative observations or one or more observations have zero values. The harmonic mean is based on the reciprocal of the numbers average. It is defined as the reciprocal of the arithmetic mean of the reciprocal of the individual observations.
Important Links:
Example 1:-
The monthly income (in rupees) of 10 persons working in a firm is as follows:
1487, 1493, 1502, 1446, 1475, 1492, 1572, 1516, 1468, 1489. Find the average monthly income.
Solution:
Let income be denoted by x.
∑ x= 1487+1493+1502+1446+1475+1492+1572+1516+1468+1489= 14,940
Bar x= ∑X/n = 14,940/10 = 1494.
(2) Median:
The median is the measure of Central Tendency which appears in the "middle" of an ordered sequence of values. That is, half of the observations in a set of data are smaller than it and half of the observations are greater than it.
As distinct from the arithmetic mean which is calculated from the value of every observation in the series, the median is what is called a 'positional' average. The term 'Position' refers to the place of value in a series. The place of the median in a series is such that an equal number of observations lie on either side of it. Median is thus the central value of the distribution of the value that divides the distribution into two equal parts. If there are even number of observations in a series, there is no actual value exactly in the middle of the series and as such the median is indeterminate. In his case the median is arbitrarily taken to be half-way between the two middle observations.
(3) Mode:
Mode is the most typical or commonly observed value in a set of data. For example, if we take the values of six different observations as 5, 8, 10, 8, 5, 8 mode will be 8, as it has occurred maximum number of times, i.e., 3 times. Graphically, it is the value on the X-axis below the peak, or highest point, of the frequency curve as can be seen from the following.
There are many situations in which arithmetic mean and median fail to reveal the true characteristic of data. For example, when we talk of most common wage, most common income, most common height, we have in mind mode and not the arithmetic mean or median discussed earlier. The mean does not always provide an accurate reflection of the data due to the presence of extreme values. Median may also prove to be quite unrepresentative of the data owing to an uneven distribution of the series. Both these shortcoming may be overcome by the use of mode. Mode refers to that value which occurs most frequently in a distribution. Mode is the easiest to compute since it is the value corresponding to the highest frequency.
(4) Geometric Mean:
Geometric Mean is defined as the N root of the product of N observations of a series. If there are two observations, we take the square root, if there are three observations, the cube root; and so on.
Symbolically, ∑ log x = 10.1259
Where X1, X2, ..........Xn refer to the various observations of the series.
(5) Harmonic Mean:
The Harmonic Mean, like the Arithmetic and Geometric mean, is computed from all observations. It is useful in special cases for averaging rates. However, Harmonic Mean cannot be computed when there are both positive and negative observations or one or more observations have zero values. The harmonic mean is based on the reciprocal of the numbers average. It is defined as the reciprocal of the arithmetic mean of the reciprocal of the individual observations.
Important Links:
- Averages: Mixtures Allegations Notes for Competitive Exams
- Arithmetic Practice Paper for Competitive Exams
- Marketing Aptitude Practice Paper for Bank Tests
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