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Mean

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Introduction to Mean

Imagine you and four friends scored marks in a test: some scored higher, some scored lower. You want to find a single number that best represents everyone's performance. This number is called the mean, commonly known as the average.

The mean gives us a central value summarising a group of numbers. It helps answer questions like, "On average, how well did the class perform?" or "What is the typical price of these items?"

In competitive exams, understanding mean is crucial because it frequently appears in questions involving scores, measurements, money, or grouped data.

Other measures such as median and mode also describe central tendency but in different ways, which we will explore later in this chapter.

Definition and Calculation of Mean

What is Mean? The mean of a set of observations is the sum of all observations divided by the total number of observations.

In simple terms, if you add up all numbers and then split this total equally among the same count of numbers, each share is the mean.

Arithmetic Mean

\[\bar{x} = \frac{\sum_{i=1}^n x_i}{n}\]

Sum of all observations divided by number of observations

\(x_i\) = Each observation
n = Number of observations
\(\bar{x}\) = Mean value

Let's see this visually:

Data set: 5, 7, 8, 10, 10 5 7 8 10 10 Sum = 5 + 7 + 8 + 10 + 10 = 40 Mean = Total Sum / Number of Observations = 40/5 = 8

Therefore, the mean of this data set is 8.

Weighted Mean

Sometimes, not all observations have equal importance. For example, if you want to find the average price per kilogram of rice bought in different quantities at different rates, simply averaging the prices without considering quantity (the weight) will give the wrong answer.

Here, more quantity means more weightage to the corresponding price. This is called the weighted mean.

The formula for weighted mean is:

Weighted Mean

\[\bar{x} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}\]

Sum of observed values multiplied by weights divided by sum of weights

\(x_i\) = Observations
\(w_i\) = Weights corresponding to observations
\(\bar{x}\) = Weighted mean

Consider this table showing quantities of rice bought at different prices:

Rice Prices and Quantities
Price per kg (Rs.) Quantity (kg)
40 5
45 3
50 2

Calculating the weighted mean price per kg:

  • Multiply each price by quantity: \(40 \times 5 = 200\), \(45 \times 3 = 135\), \(50 \times 2 = 100\)
  • Add these products: \(200 + 135 + 100 = 435\)
  • Add total quantity: \(5 + 3 + 2 = 10\) kg
  • Weighted mean price = \(\frac{435}{10} = 43.5\) Rs. per kg

Short-cut Method for Mean (Assumed Mean Method)

When the numbers are large or the data is grouped, calculating mean by direct addition can be tedious. The assumed mean method simplifies calculations by choosing an assumed value close to the data's center.

Steps involved:

  • Choose an assumed mean \(A\) (any data value, preferably central)
  • Calculate deviations \(d_i = x_i - A\) for all values
  • Sum all deviations: \(\sum d_i\)
  • Apply formula:

Short-cut Mean (Assumed Mean)

\[\bar{x} = A + \frac{\sum d_i}{n}\]

Mean calculated using deviations from assumed mean

A = Assumed mean
\(d_i\) = Deviation (x_i - A)
n = Number of observations 
graph TD    A[Choose Assumed Mean A] --> B[Calculate Deviations d_i = x_i - A]    B --> C[Sum Deviations S = ∑ d_i]    C --> D[Calculate Mean as: \( \\bar{x} = A + \frac{S}{n} \)]    D --> E[Obtain Final Mean]

Worked Examples

Example 1: Calculating Mean of Simple Numbers Easy
Find the mean of the numbers 12, 15, 18, 20, and 25.

Step 1: Add the numbers: \(12 + 15 + 18 + 20 + 25 = 90\)

Step 2: Count the numbers: there are 5 in total.

Step 3: Use the mean formula:

\(\bar{x} = \frac{90}{5} = 18\)

Answer: The mean is 18.

Example 2: Weighted Mean Using Prices and Quantities Medium
A shop sold 5 kg of rice at Rs.200 and 3 kg at Rs.180. Find the average price per kg of rice.

Step 1: Calculate total cost for each quantity:

  • 5 kg at Rs.200 total cost: \(5 \times 200 = 1000\)
  • 3 kg at Rs.180 total cost: \(3 \times 180 = 540\)

Step 2: Add total costs: \(1000 + 540 = 1540\)

Step 3: Add total quantities: \(5 + 3 = 8\) kg

Step 4: Average price per kg = \(\frac{1540}{8} = 192.5\) Rs.

Answer: The average price per kg is Rs.192.50.

Example 3: Short-cut Method with Large Data Set Medium
Calculate the mean of 42, 45, 48, 50, and 53 using the assumed mean method.

Step 1: Choose an assumed mean, say \(A = 48\) (close to middle values).

Step 2: Calculate deviations \(d_i = x_i - A\):

  • \(42 - 48 = -6\)
  • \(45 - 48 = -3\)
  • \(48 - 48 = 0\)
  • \(50 - 48 = 2\)
  • \(53 - 48 = 5\)

Step 3: Sum the deviations: \(-6 - 3 + 0 + 2 + 5 = -2\)

Step 4: Number of observations \(n = 5\).

Step 5: Apply formula:

\(\bar{x} = 48 + \frac{-2}{5} = 48 - 0.4 = 47.6\)

Answer: The mean is 47.6.

Example 4: Mean from Grouped Frequency Distribution Hard
Find the mean of the following grouped data:
Class IntervalFrequency
10 - 205
20 - 308
30 - 407

Step 1: Find mid-points of each class interval \(x_i\):

  • 10-20: \(\frac{10 + 20}{2} = 15\)
  • 20-30: \(\frac{20 + 30}{2} = 25\)
  • 30-40: \(\frac{30 + 40}{2} = 35\)

Step 2: Multiply mid-points by frequencies (\(f_i \times x_i\)):

  • 15 x 5 = 75
  • 25 x 8 = 200
  • 35 x 7 = 245

Step 3: Sum frequencies = \(5 + 8 + 7 = 20\)

Step 4: Sum products = \(75 + 200 + 245 = 520\)

Step 5: Apply mean formula for grouped data:

\(\bar{x} = \frac{520}{20} = 26\)

Answer: The mean is 26.

Example 5: Mean with Mixed Metric Units Hard
A car travels 60 km in 1 hour and 30000 meters in 30 minutes. Calculate the average speed in km/h.

Step 1: Convert meters to kilometers: \(30000\,m = \frac{30000}{1000} = 30\,km\)

Step 2: Total distance travelled = \(60 + 30 = 90\) km

Step 3: Total time taken: \(1\,hr + 0.5\,hr = 1.5\) hours

Step 4: Average speed = \(\frac{\text{Total distance}}{\text{Total time}} = \frac{90}{1.5} = 60\) km/h

Answer: The average speed is 60 km/h.

Formula Bank

Mean (Arithmetic Mean)
\[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \]
where: \(x_i\) = each observation, \(n\) = total observations, \(\bar{x}\) = mean
Weighted Mean
\[ \bar{x} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} \]
where: \(x_i\) = observations, \(w_i\) = weights, \(\bar{x}\) = weighted mean
Short-cut Method (Assumed Mean)
\[ \bar{x} = A + \frac{\sum d_i}{n} \]
where: \(A\) = assumed mean, \(d_i = x_i - A\) = deviations, \(n\) = number of observations

Tips & Tricks

Tip: Use the assumed mean method to reduce large numbers and simplify addition.

When to use: For large or grouped data sets to save time.

Tip: Always convert all units to the same system before calculating mean in mixed-unit problems.

When to use: For problems involving different metric units like km and meters.

Tip: In weighted mean problems, carefully multiply values by their weights before adding.

When to use: In price-quantity or marks-based weighted average questions.

Tip: Use mid-points for grouped data when calculating mean.

When to use: For frequency class intervals instead of direct values.

Tip: Confirm the question requires simple mean or weighted mean before starting calculations.

When to use: To apply correct formula and avoid mistakes.

Common Mistakes to Avoid

❌ Summing values but dividing by incorrect number of observations.
✓ Always count all data points and frequencies correctly before division.
Why: Miscounting leads to wrong mean values.
❌ Ignoring weights in weighted mean calculations.
✓ Multiply each value by its corresponding weight before summing.
Why: Treated data as equally weighted causes errors.
❌ Mixing units (e.g., km and meters) without conversion.
✓ Convert all measurements to the same unit before calculating mean.
Why: Different units distort average calculations.
❌ Forgetting to add the average deviation to the assumed mean in the short-cut method.
✓ Use \(\bar{x} = A + \frac{\sum d_i}{n}\), combining both parts correctly.
Why: Students stop at calculating deviations, missing final mean.
❌ Using class intervals directly without computing mid-points for grouped data mean.
✓ Calculate mid-points and use them as representative values.
Why: Class intervals are ranges, and mid-points properly estimate data location.
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