What is Simple Arithmetic Mean?
Last Updated :
03 Jul, 2023
Arithmetic Mean is one approach to measure central tendency in statistics. This measure of central tendency involves the condensation of a huge amount of data to a single value. Arithmetic mean can be determined using two methods; viz., Simple Arithmetic Mean and Weighted Arithmetic Mean.
Meaning of Simple Arithmetic Mean
Simple Arithmetic Mean is the sum of a set of numbers divided by the total number of values. It is also referred to as the average. For instance, if there are four items in a series, i.e. 6, 7, 8, and 9. The simple arithmetic mean is (6 + 7 + 8 + 9) / 4 = 7.5.
The calculation of the arithmetic mean can be done in the following series:
- Individual Series;
- Discrete Series; and
- Continuous Series
(1) Individual Series
The series in which the items are listed singly is known as Individual Series. In simple terms, a separate value of the measurement is given to each item. For example, if the weight of 10 students in a class is given individually, then the resultant series will be an individual series.
Arithmetic Mean in Individual Series
The mean is calculated by adding up all the observations and dividing it by the total number of observations in the set. There are three ways to determine the arithmetic mean of an individual series:
- Direct Method;
- Short-Cut Method; and
- Step Deviation Method.
1. Direct Method
This method involves adding up all the units and dividing the total by the number of items. The resultant quotient becomes the arithmetic mean.
Steps of Direct Method:
1. Assume that there are X1, X2,..................Xn items (observations).
2. Determine the sum, or ΣX, by adding the values of all the items.
3. Determine N; i.e., the total number of items in the series.
4. Finally, to get the mean value, divide the total value of all items (ΣX) by the total number of items (N), i.e.
\bar{X}=\frac{\sum{X}}{N}
{Where, X̄ = Arithmetic Mean; ΣX = Sum of all the values of items; N = Total number of items}
Example: Calculate the arithmetic mean of the weight of 5 students in a class by using Direct Method.
Solution:
Total Weight (ΣX) = 225, Total number of students (N) = 5
\bar{X}=\frac{\sum{X}}{N}
\bar{X}=\frac{225}{5}
Arithmetic Mean = 45 kg
2. Short-Cut Method
This method involves assuming any figure of the data set as mean and calculating deviation on its basis. When there are a large number of observations or when it becomes difficult to calculate the arithmetic mean by the direct method, then the Short-Cut method is used. This method is also known as the Assumed Mean Method.
Steps of Short-Cut Method
1. Assume that there are X1, X2,..................Xn items (observations).
2. Select an item of the series as the Assumed Mean (A).
3. Determine the series' deviations (d) from the assumed mean (A); i.e., subtract A from each item of the series to get X - A.
4. Write the total number of deviations and denote it as Σd.
5. Determine N, which implies the total number of items in the series.
6. Finally, use the formula below to calculate the mean value:
\bar{X}=A+\frac{\sum{d}}{N}
{X̄ = Arithmetic Mean; A = Assumed Mean; d = X - A; i.e., deviations of variables from assumed mean; Σd = Σ(X-A); i.e., a sum of deviations of variables from assumed mean; N =Total number of items}
Example: Calculate the arithmetic mean of the weight of the 5 students in the class by using the assumed mean method.
Solution:
\bar{X}=A+\frac{\sum{d}}{N}
\bar{X}=40+\frac{25}{5}
= 40 + 5
Arithmetic Mean = 45 kg
3. Step Deviation Method
The short-cut method was made simpler by the step deviation method. The deviations are calculated using this method by dividing deviations from the assumed mean by a common factor (C). Then, the value of the arithmetic mean is then calculated by using these step deviations.
Steps of the Step Deviation Method
1. Assume that the items (observations) are X1, X2,...................Xn.
2. Select one item in the series as the assumed mean (A).
3. Subtract the assumed mean (A) from each item in the series to determine the deviations (d) of those items from that mean.
4. Determine the common factor (C) from d and use that value to calculate d′ (step deviations), which equals \frac{d}{C} .
5. Denote the sum of step deviations (d′) as Σd′.
6. Determine N, which is the total number of items in the series.
7. Finally apply the following formula to get the value of the arithmetic mean:
\bar{X}=A+\frac{\sum{d'}}{N}\times{C}
{Where, X̄ = Arithmetic Mean; A = Assumed Mean; d = X - A, i.e., Deviations of variables from Assumed Mean; d'=\frac{X-A}{C} ; i.e., Step Deviations (deviations divided by common factor); Σd' = Sum of Step Deviations; C = Common Factor; N = Total number of items}
Example: Calculate the arithmetic mean of the weight of the 5 students in the class by using the step deviation method.
Solution:
\bar{X}=A+\frac{\sum{d'}}{N}\times{C}
\bar{X}=40+\frac{5}{5}\times{5}
= 40 + 5
Arithmetic Mean = 45
(2) Discrete Series
In the case of discrete series (ungrouped frequency distribution), the values of variables represent the repetitions. It means that the frequencies are given corresponding to the different values of variables. The total number of observations in a discrete series, N, equals the sum of the frequencies, which is Σf.
Arithmetic mean in discrete series can be calculated by using:
- Direct Method;
- Short-Cut Method; and
- Step Deviation Method.
1. Direct Method
Simple arithmetic mean can be calculated with the direct method by multiplying every item (X) by their corresponding frequencies (f), then dividing the sum of products (fX) by the total number of frequencies (f).
\bar{X}=\frac{\sum{fX}}{\sum{f}}
Steps of Direct Method
1. Multiply the various values of the variables (X) by the frequency of each variable (f), and then represent the result as fX.
2. Find the sum of fX and signify it with ΣfX.
3. Determine the series' total number of items, which is Σf or N.
4. Finally apply the following formula:
\bar{X}=\frac{\sum{fX}}{\sum{f}}
{Where, X̄ = Arithmetic Mean; ΣfX = Sum of the product of variables with their respective frequencies; Σf = Total number of items}.
Example: Calculate the arithmetic mean of the wages (per day) of 10 workers in a factory.
Solution:
\bar{X}=\frac{\sum{fX}}{\sum{f}}
\bar{X}=\frac{2050}{10}
Arithmetic Mean = ₹205 per day
2. Short-Cut Method
The mean in discrete series can also be determined using the shortcut method. Using this method, the calculation of the mean takes much less time.
Steps in Short-cut Method
1. Denote X for the variable and f for the frequency.
2. Select an item from the series as the assumed mean (A).
3. Calculate (X - A) for each item in the series to get the deviations (d) of the items.
4. Multiply the deviations (d) by the corresponding frequency (f), then add the results together to get Σfd.
5. Determine the total number of items in the series; i.e., Σf or N.
6. Use the formula below:
\bar{X}=A+\frac{\sum{fd}}{\sum{f}}
{Where, X̄ = Arithmetic Mean; A = Assumed Mean; d = X - A; i.e., deviations of variables from assumed mean; Σfd = sum of the product of deviations of variables with their respective frequencies; Σf = Total number of items}.
Example: Calculate the arithmetic mean of the wages(per day) of 10 workers in the factory using the short-cut method.
Solution:
\bar{X}=A+\frac{\sum{fd}}{\sum{f}}
\bar{X}=200+\frac{50}{10}
Arithmetic Mean = ₹205 per day
3. Step Deviation Method
This method simplifies the short-cut method even further. Under this method, to simplify the calculation, the values of the deviations (d) are divided by a common factor (C). Step deviations are the deviations that arise from this division.
Steps in Step Deviation Method
1. Write X for the variable and f for the frequency.
2. Select an item from the series as the assumed mean (A).
3. Calculate (X - A) for each item in the series to get the deviations (d) of the items.
4. Determine the common factor (C) from d, and then calculate d′ (step deviations), which is \frac{d}{C} .
5. Multiply the step deviations (d′) with the frequency (f), then add the results together to get Σfd'.
6. Determine the total number of items in the series, which is Σf or N.
7. Lastly, apply the following formula:
\bar{X}=A+\frac{\sum{fd'}}{\sum{f}}\times{C}
{Where, X̄ = Arithmetic Mean; A = Assumed Mean; d= X - A; i.e., Deviations of variables from Assumed Mean; d' = Step Deviations (deviations divided by common factor); Σfd' = Sum of product of step deviations (d') with their respective frequencies; C = Common Factor; Σf =Total number of items}
Example: Calculate the arithmetic mean of the wages(per day) of 10 workers in the factory using the step deviation method.
Solution:
\bar{X}=A+\frac{\sum{fd'}}{\sum{f}}\times{C}
\bar{X}=200+\frac{5}{10}\times{10}
Arithmetic Mean = ₹205 per day
(3) Continuous Series
In continuous series (grouped frequency distribution), the value of a variable is grouped into several class intervals (such as 0-5,5-10,10-15) along with the corresponding frequencies. The method used to determine the arithmetic average in a continuous series is the same as that used in discrete series. The midpoints of several class intervals replace the class interval in a continuous series. When it is done, a continuous series and a discrete series are the same.
The arithmetic mean in continuous series can be calculated by using:
- Direct Method;
- Shortcut Method; and
- Step Deviation Method.
1. Direct Method
The direct method for continuous series is the same as for discrete series, with the difference that the continuous series is changed into a discrete series by determining the midpoints of the class interval.
Steps of the Direct Method
1. Determine each class interval's midpoint and represent it by the symbol m.
2. Multiply the mid-points (m) by their respective frequencies (f) to get fm.
3. Determine the total of fm and represent it as ∑fm.
4. Figure out the total number of items in the series; i.e., Σf or N.
5. Use the formula below:
\bar{X}=\frac{\sum{fm}}{\sum{f}}
{Where, X̄ = Arithmetic Mean; Σfm = Sum of the product of mid-points with their respective frequencies; Σf = Total number of items}
Example: Calculate the arithmetic mean of the following data.
Solution:
\bar{X}=\frac{\sum{fm}}{\sum{f}}
\bar{X}=\frac{560}{20}
Arithmetic Mean = 28
2. Short-cut Method
In the case of continuous series, the shortcut approach saves a significant amount of time when calculating the mean.
Steps in Short-cut Method
1. Determine the mid-points of each class interval and indicate them with the letter m.
2. Choose one mid-point as the assumed mean (A).
3. Determine the deviations (d) of mid-points from the assumed mean (A); i.e., calculate (m - A).
4. Multiply the deviations (d) by the corresponding frequency (f), then add them together to get ∑fd.
5. Figure out the total number of items in the series; i.e., Σf or N.
6. Use the formula below:
\bar{X}=A+\frac{\sum{fd}}{\sum{f}}
{Where, X̄ = Arithmetic Mean; A = Assumed Mean; d = m - A; i.e., deviations of mid-points from assumed mean; Σfd = sum of the product of deviations with their respective frequencies; Σf = Total number of items}.
Example: Calculate the arithmetic mean of the following data using the short-cut method.
Solution:
\bar{X}=A+\frac{\sum{fd}}{\sum{f}}
\bar{X}=25+\frac{60}{20}
Arithmetic Mean = 28
3. Step-Deviation Method
When all of the class intervals in a continuous series have the same magnitude (width), the step-deviation method can further simplify the short-cut method.
Steps in Step-Deviation Method
1. Determine the mid-points of each class interval and indicate them with the letter m.
2. Choose one mid-point as the assumed mean (A).
3. Determine the deviations (d) of mid-points from the assumed mean (A); i.e., calculate (m - A).
4. Determine the common factor (C) from d, and then calculate d′ (step deviations), which is \frac{d}{C} .
5. Multiply the step deviations (d′) with the frequency (f), then add the results together to get Σfd'.
6. Figure out the total number of items in the series; i.e., Σf or N.
7. Use the formula below:
\bar{X}=A+\frac{\sum{fd'}}{\sum{f}}\times{C}
{Where, X̄ = Arithmetic Mean; A = Assumed Mean; d = m - A; i.e., deviations of mid-points from assumed mean; d'= Step Deviations (deviations divided by common factor); Σfd' = Sum of product of step deviations (d') with their respective frequencies (f); C = Common Factor; Σf =Total number of items}.
Example: Calculate the arithmetic mean of the following data using the Step-Deviation method.
Solution:
\bar{X}=A+\frac{\sum{fd'}}{\sum{f}}\times{C}
\bar{X}=25+\frac{6}{20}\times{10}
Arithmetic Mean = 28
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Calculation of Mode in Discrete Series | Formula of ModeThe word mode comes from the Latin word 'Modus', meaning measurements, quantity, way, or manner. In statistics, Mode refers to the variable that occurs most of the time or repeats itself most frequently in a given series of variables (say X). It is a maximum occurrence at a particular point or a val
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Grouping Method of Calculating Mode in Discrete Series | Formula of ModeMode is the value from a data set that has occurred the most number of times. It is one of the most crucial measures of central tendency and is represented by Z. For example, if, in a class of 35 students, 10 students are 10 years old, 20 students are 11 years old, and the rest 5 students are 9 year
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Calculation of Mode in Continuous Series | Formula of ModeMode, in statistics, refers to the variable that occurs most of the time in the given series. In simple words, a mode is a variable that repeats itself most frequently in a given series of variables (say X). We can determine the mode in two series; viz., individual and discrete series. Mode is denot
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Calculation of Mode in Special CasesThe word mode is derived from the French word âLa Modeâ, meaning anything that is in fashion or vogue. A measure of central tendency in statistical series that determines the value occurring most frequently in the given series is known as mode. In other words, the modal value of the series has the h
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Calculation of Mode by Graphical MethodThe word mode is derived from the French word 'La Mode', meaning anything that is in fashion or vogue. A measure of central tendency in statistical series that determines the value occurring most frequently in the given series is known as mode. In other words, the modal value of the series has the h
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Mean, Median and Mode| Comparison, Relationship and CalculationA single value used to symbolise a whole set of data is called the Measure of Central Tendency. In comparison to other values, it is a typical value to which the majority of observations are closer. Average and Measure of Location are other names for the Measure of Central Tendency. In statistical a
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Chapter 8: Measures of Dispersion
Measures of Dispersion | Meaning, Absolute and Relative Measures of DispersionAverages like mean, median, and mode can be used to represent any series by a single number. However, averages are not enough to describe the characteristics of statistical data. So, it is necessary to define some additional summary measures to adequately represent the characteristics of a distribut
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Range | Meaning, Coefficient of Range, Merits and Demerits, Calculation of RangeWhat is Range?Range is the easiest to understand of all the measures of dispersion. The difference between the largest and smallest item in a distribution is called range. It can be written as: Range (R) = Largest item (L) â Smallest item (S) For example, If the marks of 5 students of class Xth are
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Calculation of Range and Coefficient of RangeWhat is Range?Range is the easiest to understand of all the measures of dispersion. The difference between the largest and smallest item in a distribution is called range. It can be written as:Range (R) = Largest item (L) â Smallest item (S)For example, If the marks of 5 students of class XIth are 2
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Interquartile Range and Quartile DeviationThe extent to which the values of a distribution differ from the average of that distribution is known as Dispersion. The measures of dispersion can be either absolute or relative. The Measures of Absolute Dispersion consist of Range, Quartile Deviation, Mean Deviation, Standard Deviation, and Loren
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Partition Value | Quartiles, Deciles and PercentilesPartition values are statistical measures that divide a dataset into equal parts. They help in understanding the distribution and spread of data by indicating where certain percentages of the data fall. The most commonly used partition values are quartiles, deciles, and percentiles.Table of ContentW
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Quartile Deviation and Coefficient of Quartile Deviation: Meaning, Formula, Calculation, and ExamplesThe extent to which the values of a distribution differ from the average of that distribution is known as Dispersion. The measures of dispersion can be either absolute or relative. The Measures of Absolute Dispersion consist of Range, Quartile Deviation, Mean Deviation, Standard Deviation, and Loren
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Quartile Deviation in Discrete Series | Formula, Calculation and ExamplesWhat is Quartile Deviation?Quartile Deviation (absolute measure) divides the distribution into multiple quarters. Quartile Deviation is calculated as the average of the difference of the upper quartile (Q3) and the lower quartile (Q1).Quartile~Deviation=\frac{Q_3-Q_1}{2} Where,Q3 = Upper Quartile (S
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Quartile Deviation in Continuous Series | Formula, Calculation and ExamplesWhat is Quartile Deviation?Quartile Deviation (absolute measure) divides the distribution into multiple quarters. Quartile Deviation is calculated as the average of the difference of the upper quartile (Q3) and the lower quartile (Q1).Quartile~Deviation=\frac{Q_3-Q_1}{2} Where,Q3 = Upper Quartile (S
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Mean Deviation: Coefficient of Mean Deviation, Merits, and DemeritsRange, Interquartile range, and Quartile deviation all have the same defect; i.e., they are determined by considering only two values of a series: either the extreme values (as in range) or the values of the quartiles (as in quartile deviation). This approach of analysing dispersion by determining t
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Calculation of Mean Deviation for different types of Statistical SeriesWhat is Mean Deviation?The arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode) is known as the Mean Deviation of a series. Other names for Mean Deviation are the First Moment of Dispersion and Average Deviation. Mean deviation is calculate
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Mean Deviation from Mean | Individual, Discrete, and Continuous SeriesMean Deviation of a series can be defined as the arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode). Mean Deviation is also known as the First Moment of Dispersion or Average Deviation. Mean Deviation is based on all the items of the seri
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Mean Deviation from Median | Individual, Discrete, and Continuous SeriesWhat is Mean Deviation from Median?Mean Deviation of a series can be defined as the arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode). Mean Deviation is also known as the First Moment of Dispersion or Average Deviation. Mean Deviation is
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Standard Deviation: Meaning, Coefficient of Standard Deviation, Merits, and DemeritsThe methods of measuring dispersion such as quartile deviation, range, mean deviation, etc., are not universally adopted as they do not provide much accuracy. Range does not provide required satisfaction as in the entire group, range's magnitude is determined by most extreme cases. Quartile Deviatio
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Standard Deviation in Individual SeriesA scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol Ï (sigma). Under this method, the deviation of valu
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Standard Deviation in Discrete SeriesA scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol Ï (sigma). Under this method, the deviation of valu
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Standard Deviation in Frequency Distribution SeriesA scientific measure of dispersion that is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. It is denoted by a Greek Symbol Ï (sigma). Under this method, the deviation of values is taken from
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Combined Standard Deviation: Meaning, Formula, and ExampleA scientific measure of dispersion, which is widely used in statistical analysis of a given set of data is known as Standard Deviation. Another name for standard deviation is Root Mean Square Deviation. Standard Deviation is denoted by a Greek Symbol Ï (sigma). Under this method, the deviation of va
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Coefficient of Variation: Meaning, Formula and ExamplesWhat is Coefficient of Variation? As Standard Deviation is an absolute measure of dispersion, one cannot use it for comparing the variability of two or more series when they are expressed in different units. Therefore, in order to compare the variability of two or more series with different units it
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Lorenz Curveb : Meaning, Construction, and ApplicationWhat is Lorenz Curve?The variability of a statistical series can be measured through different measures, Lorenz Curve is one of them. It is a Cumulative Percentage Curve and was first used by Max Lorenz. Generally, Lorenz Curves are used to measure the variability of the distribution of income and w
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Chapter 9: Correlation