Mean of Grouped Data

Definition: The mean (or arithmetic mean) of grouped data is the average value, calculated by dividing the sum of all values by the total number of observations. Since individual values are unknown in grouped data, class marks are used as representative values.

Key terms:

• Class interval — a range of values (e.g., 10-20, 20-30).
• Class mark (x_i) — midpoint of a class interval = (upper limit + lower limit)/2.
• Frequency (f_i) — number of observations in a class interval.
• Sigma f_i — total number of observations (sum of all frequencies).
• Assumed mean (a) — a convenient value (usually the class mark of the middle class) used to simplify calculations.
• Deviation (d_i) — d_i = x_i - a (difference of class mark from assumed mean).
• Step-deviation (u_i) — u_i = (x_i - a)/h, where h is the class size.

Derivation of the Assumed Mean Method:

1. Let the true mean be x-bar = Sigma(f_i x_i) / Sigma(f_i).
2. Let a be the assumed mean. Define d_i = x_i - a, so x_i = a + d_i.
3. Sigma(f_i x_i) = Sigma f_i(a + d_i) = a x Sigma(f_i) + Sigma(f_i d_i).
4. x-bar = a + Sigma(f_i d_i) / Sigma(f_i).
5. This is the Assumed Mean formula. Hence derived.

Derivation of the Step-Deviation Method:

1. Start from d_i = x_i - a.
2. If all class sizes are equal (= h), then d_i is always a multiple of h.
3. Define u_i = d_i / h = (x_i - a)/h. So d_i = h x u_i.
4. Sigma(f_i d_i) = Sigma f_i(h x u_i) = h x Sigma(f_i u_i).
5. x-bar = a + h x Sigma(f_i u_i) / Sigma(f_i).
6. This is the Step-Deviation formula. Hence derived.

All three methods are algebraically equivalent and always produce the same answer.

Problems on mean of grouped data in Class 10 include:

Type 1: Calculate Mean Using Direct Method

• Construct the f_i x x_i column, sum it, and divide by Sigma(f_i).

Type 2: Calculate Mean Using Assumed Mean Method

• Choose a suitable assumed mean (a), calculate d_i = x_i - a, find Sigma(f_i d_i).

Type 3: Calculate Mean Using Step-Deviation Method

• Choose a, calculate u_i = (x_i - a)/h, find Sigma(f_i u_i).

Type 4: Finding Missing Frequency

• Given the mean and all but one frequency, find the missing frequency using the mean formula.

Type 5: Combined Mean

• Combine two groups with known means and total frequencies to find the overall mean.

Type 6: Comparison of Methods

• Solve the same problem using all three methods and verify the same answer.

Direct Method — Step by Step:

1. Write the class intervals and their frequencies (f_i).
2. Find the class mark for each interval: x_i = (upper limit + lower limit)/2.
3. Calculate f_i x x_i for each class.
4. Sum all f_i to get Sigma(f_i) = N.
5. Sum all f_i x x_i to get Sigma(f_i x_i).
6. Mean = Sigma(f_i x_i) / Sigma(f_i).

Assumed Mean Method — Step by Step:

1. Find class marks (x_i).
2. Choose an assumed mean (a) — usually the class mark of the middle class or highest-frequency class.
3. Calculate d_i = x_i - a for each class.
4. Calculate f_i x d_i for each class.
5. Sum to get Sigma(f_i d_i).
6. Mean = a + Sigma(f_i d_i) / Sigma(f_i).

Step-Deviation Method — Step by Step:

1. Find class marks (x_i) and class size h.
2. Choose an assumed mean (a).
3. Calculate u_i = (x_i - a)/h for each class.
4. Calculate f_i x u_i for each class.
5. Sum to get Sigma(f_i u_i).
6. Mean = a + [Sigma(f_i u_i) / Sigma(f_i)] x h.

Tips:

• Choose the assumed mean close to the expected mean (usually the middle class mark).
• The step-deviation method works only when class sizes are equal.
• Always verify: the mean should lie within the range of the data.

Economics:

• Calculating average income, expenditure, or GDP growth from grouped data collected across income brackets.

Education:

• Finding average marks of students when marks are grouped in intervals (0-20, 20-40, etc.).

Demographics:

• Census data uses grouped frequency distributions. Mean calculations provide summary statistics for age, household size, etc.

Quality Control:

• Manufacturing data on product weights/dimensions is grouped. Mean calculations check if production meets specifications.

Healthcare:

• Patient data (blood pressure, BMI) is grouped. Mean values help assess population health trends.

Government:

• Grouped data on rainfall, temperature, or crop yield is summarised using means for policy decisions.