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How Quantities Combine: Understanding Data - Combining Things

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The process of combining quantities in data analysis often involves finding a representative value, such as the Arithmetic Mean, which accounts for every observation in the data set.

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The Arithmetic Mean (or simply Mean) of a set of observations is the sum of the values of all the observations divided by the total number of observations.

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When data is presented in a frequency distribution table, where observations x1,x2,…,xnx_1, x_2, \dots, x_n occur with frequencies f1,f2,…,fnf_1, f_2, \dots, f_n, the quantities are combined using the weighted sum ∑fixi\sum f_i x_i.

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Combining two groups: If one group of n1n_1 items has a mean xˉ1\bar{x}_1 and another group of n2n_2 items has a mean xˉ2\bar{x}_2, the combined mean xˉc\bar{x}_c is calculated by finding the total sum of all items across both groups.

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The Range of a data set is the difference between the maximum and minimum values, representing the spread of the combined quantities: Range=Xmax−XminRange = X_{max} - X_{min}.

📐Formulae

xˉ=∑i=1nxin\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

xˉ=f1x1+f2x2+⋯+fkxkf1+f2+⋯+fk=∑i=1kfixi∑i=1kfi\bar{x} = \frac{f_1 x_1 + f_2 x_2 + \dots + f_k x_k}{f_1 + f_2 + \dots + f_k} = \frac{\sum_{i=1}^{k} f_i x_i}{\sum_{i=1}^{k} f_i}

xˉcombined=n1xˉ1+n2xˉ2n1+n2\bar{x}_{combined} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

💡Examples

Problem 1:

The mean marks obtained by a class of 3030 students is 7070 and the mean marks of another class of 2020 students is 8080. Find the combined mean of the marks of all 5050 students.

Solution:

Given: Number of students in Class 1 (n1n_1) = 3030 Mean of Class 1 (xˉ1\bar{x}_1) = 7070 Number of students in Class 2 (n2n_2) = 2020 Mean of Class 2 (xˉ2\bar{x}_2) = 8080

Step 1: Calculate the total marks for Class 1. Total marks1=n1×xˉ1=30×70=2100\text{Total marks}_1 = n_1 \times \bar{x}_1 = 30 \times 70 = 2100

Step 2: Calculate the total marks for Class 2. Total marks2=n2×xˉ2=20×80=1600\text{Total marks}_2 = n_2 \times \bar{x}_2 = 20 \times 80 = 1600

Step 3: Combine the totals and the number of students. Combined Total Marks=2100+1600=3700\text{Combined Total Marks} = 2100 + 1600 = 3700 Total Students=30+20=50\text{Total Students} = 30 + 20 = 50

Step 4: Use the combined mean formula. xˉcombined=370050=74\bar{x}_{combined} = \frac{3700}{50} = 74

Therefore, the combined mean is 7474.

Explanation:

To combine the means of two different groups, we cannot simply average the means. We must find the total sum of all values (marks) from both groups and divide by the total number of items (students).

Problem 2:

Find the mean of the following frequency distribution: Observations (xx): 5,10,15,205, 10, 15, 20 Frequency (ff): 2,3,4,12, 3, 4, 1

Solution:

We calculate ∑fixi\sum f_i x_i and ∑fi\sum f_i:

  1. For x=5,f=2:f1x1=5×2=10x=5, f=2: f_1 x_1 = 5 \times 2 = 10
  2. For x=10,f=3:f2x2=10×3=30x=10, f=3: f_2 x_2 = 10 \times 3 = 30
  3. For x=15,f=4:f3x3=15×4=60x=15, f=4: f_3 x_3 = 15 \times 4 = 60
  4. For x=20,f=1:f4x4=20×1=20x=20, f=1: f_4 x_4 = 20 \times 1 = 20

Sum of fixif_i x_i: 103060+20120\begin{array}{r} 10 \\ 30 \\ 60 \\ + 20 \\ \hline 120 \end{array}

Sum of fif_i: 234+110\begin{array}{r} 2 \\ 3 \\ 4 \\ + 1 \\ \hline 10 \end{array}

Mean (xˉ\bar{x}): xˉ=∑fixi∑fi=12010=12\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{120}{10} = 12

The mean of the distribution is 1212.

Explanation:

When quantities repeat, we multiply each quantity by its frequency to find the total sum before dividing by the total number of occurrences.