Statistics: Mean, Mode, and Median Calculations

Data on Ages of Children in a School

Problem Statement

• Compute the mean age using direct and short-cut methods.
• Provisional mean suggested: $A = 8$.
• Data provided includes class limits and corresponding frequencies.

Class Distribution for Ages

• Class Limits:
  • 4-6 years: Frequency = 10
  • 7-9 years: Frequency = 20
  • 10-12 years: Frequency = 13
  • 13-15 years: Frequency = 7
• Total Frequency = 50

Calculation of Mean Age

1. Direct Method

• Step 1: Compute midpoints (x) for each class.

  • For class 4-6: $x_1 = \frac{4 + 6}{2} = 5$
  • For class 7-9: $x_2 = \frac{7 + 9}{2} = 8$
  • For class 10-12: $x_3 = \frac{10 + 12}{2} = 11$
  • For class 13-15: $x_4 = \frac{13 + 15}{2} = 14$

• Step 2: Multiply each midpoint by its corresponding frequency and sum the results.

  • $\Sigma fx = (5 \times 10) + (8 \times 20) + (11 \times 13) + (14 \times 7)$
  • $\Sigma fx = 50 + 160 + 143 + 98 = 451$

• Step 3: Calculate mean by dividing the total $\Sigma fx$ by the total frequency:

  • $\text{Mean} = \frac{\Sigma fx}{N} = \frac{451}{50} = 9.02$

2. Short-Cut Method

• Step 1: Calculate deviations from the provisional mean (A = 8).

  • For class 4-6: $d_1 = 5 - 8 = -3$
  • For class 7-9: $d_2 = 8 - 8 = 0$
  • For class 10-12: $d_3 = 11 - 8 = 3$
  • For class 13-15: $d_4 = 14 - 8 = 6$

• Step 2: Multiply deviations by frequency.

  • $f<em>1d</em>1 = 10 \times (-3) = -30$
  • $f<em>2d</em>2 = 20 \times 0 = 0$
  • $f<em>3d</em>3 = 13 \times 3 = 39$
  • $f<em>4d</em>4 = 7 \times 6 = 42$

• Step 3: Sum of the deviations: $\Sigma fd = -30 + 0 + 39 + 42 = 51$

• Step 4: Compute mean using the formula:

  • $\text{Mean} = A + \frac{\Sigma fd}{N} = 8 + \frac{51}{50} = 8 + 1.02 = 9.02$

Calculation of Geometric Mean (GM)

• Formula for GM is: $GM = (x<em>1 \times x</em>2 \times x<em>3 \times … \times x</em>n)^{\frac{1}{n}}$
• Using midpoints for GM calculation:
  • $GM = (5^{10} \times 8^{20} \times 11^{13} \times 14^{7})^{\frac{1}{50}}$

Calculation of Harmonic Mean (HM)

• Formula for HM is: $HM = \frac{n}{\frac{1}{x<em>1} + \frac{1}{x</em>2} + … + \frac{1}{x_n}}$
• Using midpoints for HM:
  • $HM = \frac{50}{\frac{10}{5} + \frac{20}{8} + \frac{13}{11} + \frac{7}{14}}$

Data on Number of Children in Various Families

Problem Statement

• Find the mode and median of the given data.

Data Distribution in Families

• Assume data presented includes a frequency distribution or list from which to compute the mode and median. (Specific numerical data needs to be referenced to compute these metrics, but is not given in the transcript.)

Calculation of Mode

• Mode is defined as the value that appears most frequently in a data set.
• To find mode:
  • List out the data points and identify the frequency of each point.
  • Select the point with the highest frequency.

Calculation of Median

• Median is defined as the middle value of a data set when arranged in ascending order:
• Steps to find the median:
  • Sort the data set in ascending order.
  • If the total number of observations (n) is odd, the median is the value at position $\frac{n+1}{2}$. If n is even, the median is the average of the values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.
• Apply the sorting and computation operation to find the median value.