Grade 10 Statistics Review Notes

Purpose of the Research Study

• Clearly define the intent or objective.
• Formulate a statement of purpose in the introduction section of the research.

Sources of Researchable Problems

• Researcher’s own experiences.
• Practical issues requiring solutions.
• Previous theories and past research.

Research Methodology

• Detailed explanation of procedure:
  • Instruments used in research.
  • Participants involved in the study.
• Sampling: The process of selecting participants from the target population.
• Sample: A subset of the population from which information is gathered.

Sampling Methods

• Systematic Random Sampling: Sample every 10th customer in a fast-food chain for a study.
• Multistage Sampling: Combines multiple sampling methods.
• Simple Random Sampling: Also called fish-bowl or lottery method.

Slovin’s Formula for Sample Size

• Formula: ( n = \frac{N}{1 + Ne^2} )
  • Where:
  • ( n ) = sample size
  • ( N ) = population size
  • ( e ) = error tolerance (e.g., 0.05)
• Example Calculation: For a population of 1800 and 5% margin of error:
  • ( n = \frac{1800}{1 + (1800)(0.05)^2} )
  • Simplifies to: ( n \approx 338 )

Measures of Central Tendency for Ungrouped Data

Mean

• Most commonly used measure of central tendency.
• Formula: ( \bar{X} = \frac{\text{Sum of all scores}}{\text{Number of scores}} )

Median

• Middle number in an ordered data set.
• Finding Median:
  • If odd count: select the middle number.
  • If even count: use ( \frac{\text{middle number1} + \text{middle number2}}{2} ).

Mode

• Most frequently occurring number in data.
  • Symbol: ( \hat{X} ) (X-hat).

Quartiles

• Divides data into four intervals.
• Formula: ( Q_k = \left[ \frac{k(N)}{4} \right]\text{th observation} )
• Interpretation:
  • ( Q_1 ): 25% of data are less than or equal to this value.
  • ( Q_2 ) (median): 50% of data are less than/equal to this value.
  • ( Q_3 ): 75% of data are less than or equal to this value.

Deciles

• Splits data into ten equal parts.
• Formula: ( D_k = \left[ \frac{k(N)}{10} \right]\text{th observation} )

Percentiles

• Divides data into 100 equal parts.
• Formula: ( P_k = \left[ \frac{k(N)}{100} \right]\text{th observation} )

Measures of Central Tendency for Grouped Data

Mean

• Formula: ( \bar{X} = \frac{\Sigma fixi}{N} )
  • Where ( fi ) = frequency, ( xi ) = class mark, ( N ) = total frequency.

Mode

• Formula: ( \hat{x} = L{mo} + \left( \frac{\Delta1}{\Delta1 + \Delta2} \right)i )
  • Where:
    • ( L_{mo} ) = lower boundary of modal class
    • ( \Delta1, \Delta2 ) = frequencies of adjacent classes.

Median

• Formula: ( \tilde{X} = Lm + \left( \frac{N/2 - C{fb}}{f_{q1}} \right)i )
  • Where ( L_m ) = lower class boundary of median class.

Quartiles, Deciles, Percentiles for Grouped Data

• Quartile Formula: ( Qk = LQk + \left( \frac{kN/4 - C{fb}}{f{Q_k}} \right)i )
• Decile Formula: ( Dk = LDk + \left( \frac{kN/10 - C{fb}}{f{D_k}} \right)i )
• Percentile Formula: ( Pk = LPk + \left( \frac{kN/100 - C{fb}}{f{P_k}} \right)i )

Steps to Analyze Grouped Data

1. Solve for the class.
   • Modal class = Highest frequency.
   • Median class = N/2.
   • Quartile class, Decile class, Percentile class = Formulas as outlined above.
2. Determine the lower-class boundary and cumulative frequency.
3. Substitute all the values into the appropriate formulas and compute.
4. Interpret the results accordingly.