Q.The marks scored by 5 first-year Commerce students in an Economics unit test (out of 100) are: 45, 50, 55, 60, 70. Calculate the arithmetic mean by the direct method, and verify your answer using the short-cut (assumed mean) method.
Concept understanding — Arithmetic Mean — Meaning and Methods of Computation
The arithmetic mean is the sum of all observations divided by their number. For an individual series it is computed directly as Xˉ=ΣX/N, or by the short-cut (assumed mean) method Xˉ=A+Σd/N using deviations d=X−A. For a discrete series the same two methods apply weighted by frequency: Xˉ=ΣfX/N or Xˉ=A+Σfd/N. For a continuous series, the step-deviation method is generally quickest: Xˉ=A+(Σfd′/N)×h, with d′=(m−A)/h computed from each class's mid-value m and common class width h. All valid methods on the same data give the same final mean; the mean is based on every observation and suited to further algebraic use, but is pulled noticeably by extreme values.
The arithmetic mean of an individual series is simply the sum of all values divided by how many there are.
Summing all five marks and dividing by 5 gives a mean of 56 marks.
Arithmetic Mean = 56 marks.
Step 1 — Direct method.
Xˉ=NΣX=545+50+55+60+70=5280=56
Step 2 — Verify by the short-cut (assumed mean) method. Take A=55 (one of the given values, for convenience) and find the deviation d=X−A of every item:
| X | d = X − A |
|---|---|
| 45 | −10 |
| 50 | −5 |
| 55 | 0 |
| 60 | 5 |
| 70 | 15 |
Σd=−10−5+0+5+15=5
Xˉ=A+NΣd=55+55=55+1=56
Both methods agree exactly.
Arithmetic Mean = 56 marks (confirmed by both the direct and short-cut methods).
Independent cross-check using a different assumed mean. Taking A=50 instead: deviations are −5,0,5,10,20, so Σd=30 and Xˉ=50+30/5=50+6=56 — the same answer regardless of which value is chosen as the assumed mean, confirming the arithmetic.
Forgetting to divide Σd by N before adding it back to A (i.e., writing Xˉ=A+Σd instead of A+Σd/N) is the single most common slip in the short-cut method; also, choosing an assumed mean far from the data makes the deviations larger and more error-prone to add, even though the final answer is unaffected by which value is chosen.
- CBSE 2024Set ANNUAL2 marksQ.Calculate A.M. for the data : 30, 20, 32, 16, 27
›Reveal solutionSolution
The arithmetic mean is the sum of the observations divided by their number. Here the sum is 125 and there are 5 observations, so the mean is 25.
Data
30, 20, 32, 16, 27
Method
Arithmetic Mean = (Sum of all observations) / (Number of observations)
Step 1 — Sum of observations: 30 + 20 + 32 + 16 + 27 = 125
Step 2 — Number of observations (n) = 5
Step 3 — Arithmetic Mean = 125 / 5 = 25
✓Final answerArithmetic Mean = 125 / 5 = 25.
- CBSE 2023Set ANNUAL2 marksQ.Find the A.M. for the data 30, 20, 32, 16, 27.
›Reveal solutionSolution
The arithmetic mean (A.M.) of 30, 20, 32, 16 and 27 is 25, obtained by dividing the sum of the observations (125) by the number of observations (5).
Method
The arithmetic mean by the direct method is found by dividing the sum of all the observations by the total number of observations.
Arithmetic Mean = Sum of all observations / Number of observations.
Working
Step 1 — Add the observations:
30 + 20 + 32 + 16 + 27 = 125.
Step 2 — Count the number of observations:
Number of observations = 5.
Step 3 — Divide the sum by the number of observations:
Arithmetic Mean = 125 / 5 = 25.
✓Final answerThe arithmetic mean of the data 30, 20, 32, 16 and 27 is 25 (sum 125 divided by 5 observations).
- CBSE 2020Set ANNUAL2 marksQ.Write a short note on: Arithmetic Mean.
›Reveal solutionSolution
Arithmetic mean (the simple average) is found by adding all the values in a data set and dividing the sum by the number of values. It is the most widely used measure of central tendency and represents the typical or central value of the data.
Meaning
Arithmetic mean is the value obtained by dividing the sum of all the observations by the total number of observations. It is what is ordinarily called the average.
For an ungrouped series:
Arithmetic Mean = (Sum of all the values) / (Number of values)
For a frequency distribution:
Arithmetic Mean = (Sum of f multiplied by X) / (Sum of f), where f is the frequency and X is the value.
For example, the arithmetic mean of 10, 20 and 30 is (10 + 20 + 30) / 3 = 60 / 3 = 20.
The arithmetic mean is simple to calculate and easy to understand, and it takes every value of the series into account, which is why it is the most commonly used average, though it can be unduly affected by very large or very small (extreme) values.
This concept in the AP Intermediate 1st-year Economics course aligns with the statistics content of the NCERT/CBSE curriculum.
✓Final answerArithmetic mean is the average found by adding all the values and dividing by the number of values (or sum of fX divided by sum of f for a frequency distribution); it is the most commonly used measure of central tendency.
- CBSE 2019Set ANNUAL2 marksQ.What is meant by Arithmetic Mean?
›Reveal solutionSolution
The arithmetic mean is a measure of central tendency found by dividing the sum of all the observations in a series by the number of observations. It is the most widely used average and represents the series by a single value.
Meaning
The arithmetic mean, commonly called the average, is the most important and widely used measure of central tendency. It is defined as the sum of all the values (items) in a series divided by the number of items:
Arithmetic Mean = Sum of all the observations / Total number of observations.
For example, if a student scores 40, 50 and 60 marks in three tests, the arithmetic mean is (40 + 50 + 60) divided by 3 = 150 / 3 = 50 marks.
The arithmetic mean gives a single representative value for the whole series, is easy to calculate and understand, and is based on all the observations — which is why it is the most popular average in economics and statistics.
✓Final answerThe arithmetic mean (simple average) is a measure of central tendency obtained by adding together all the values in a series and dividing the total by the number of observations (Arithmetic Mean = sum of observations ÷ number of observations). It gives a single representative value for the whole series and is the most commonly used average.
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