Statistical Methods in Quality Management

Statistical Methods in Quality Management

Statistical Methodology

• Descriptive Statistics: Methods of presenting data visually and numerically.

  • Charts, frequency distributions, and histograms organize and present data.

  • Measures of Central Tendency: Mean, median, mode.

  • Measures of Dispersion: Variance, standard deviation, range.

• Statistical Inference: Drawing conclusions about the unknown characteristics of a population based on sample data.

• Predictive Statistics: Developing predictions of future values from historical data.

Descriptive Statistics

• Population vs Sample:

  • Population: Complete set of objects of interest.

  • Sample: Subset of objects taken from the population.

• Measures of Location:

  • Mean: Population mean = ( \mu = \frac{\sum{i=1}^{N} xi}{N} ); sample mean = ( \bar{x} = \frac{\sum{i=1}^{n} xi}{n} ).

  • Median: The middle value in sorted data.

  • Mode: The most frequently occurring observation.

• Measures of Dispersion:

  • Range: ( \text{Range} = \text{MAX(data range)} - \text{MIN(data range)} ).

  • Variance:

  • Population: ( \sigma^2 = \frac{\sum{i=1}^{N}(xi - \mu)^2}{N} ).

  • Sample: ( s^2 = \frac{\sum{i=1}^{n}(xi - \bar{x})^2}{n-1} ).

  • Standard Deviation:

  • Population: ( \sigma = \sqrt{\sigma^2} ).

  • Sample: ( s = \sqrt{s^2} ).

• Proportions:

  • Proportion (P) is a fraction with a certain characteristic.

  • Sample proportion (( \hat{p} )) is used in categorical data analysis.

  • Example in Excel: ( \text{COUNTIF(range, criteria)} )

• Measures of Shape:

  • Skewness: Asymmetry of the data.

  • Coefficient of Skewness (CS):

    • CS > 1 or < -1: High skewness.

    • 0.5 to 1 or -0.5 to -1: Moderate skewness.

    • 0.5 to -0.5: Symmetrical.

  • Kurtosis: Peakedness or flatness of a histogram.

  • Coefficient of Kurtosis (CK) measures degree of kurtosis.

    • CK < 3: flat distribution.

    • CK > 3: peaked distribution.

Statistical Inference

• Sampling Distribution: Distribution of a statistic for all possible samples of a fixed size.

• Standard Error of the Mean:

  • Infinite populations: ( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} ).

  • Finite populations: ( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}} ).

• Central Limit Theorem:

  • The sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the population distribution.

• Confidence Intervals:

  • CI is an interval estimate of a population parameter indicating the probability of containing the true parameter.

  • General formula: ( \text{CI} = \bar{x} \pm z\left( \frac{\sigma}{\sqrt{n}} \right) \)

Hypothesis Testing

• Concept: Involves two contrasting propositions (hypotheses) regarding a population parameter (null hypothesis vs alternative hypothesis).

• Steps in Hypothesis Testing:

  1. Formulate the hypotheses.

  2. Select level of significance (α).

  3. Determine a decision rule.

  4. Collect data and compute test statistic.

  5. Apply decision rule and draw conclusion.

• Critical Value: Divides the sampling distribution into rejection and non-rejection regions.

• P-value: Probability of obtaining statistic value as extreme or more extreme than one observed if null hypothesis is true.

Analysis of Variance (ANOVA)

• Purpose: To test equality of means from multiple populations.

• One-way ANOVA: Compares means for different levels of one factor.

• F Statistic: Ratio of variance estimates.

• Decision Rule: If F statistic > critical value, means are likely different.

Regression and Correlation

• Regression Analysis: Models relationships between dependent and independent variables.

  • Simple Regression: One independent variable.

  • Multiple Regression: Multiple independent variables.

• Correlation: Measures linear relationship:

  • Ranges from -1 (perfect negative) to 1 (perfect positive), with 0 indicating no relationship.

  • Coefficient of Determination (R²): Proportion of variance in dependent variable explained by independent variable(s).

Design of Experiments

• Purpose: To test and compare methods or optimize process outputs by examining the effect of factors.

• Factorial Experiment: All combinations of factor levels included.

• Main Effect: Difference a factor has on the response.

• Interaction Effects: The effect of changing one factor on others.