Data Analysis Methods: Quantitative

Data Analysis Methods: Quantitative

By KHO CHUNG WEI

In this topic, you will:

• Analyse quantitative data using descriptive statistics

• Analyse quantitative data using inferential statistics

• ​Interpret the data

• Present the data​

TSLB3143

Fundamentals of Research in Education

Scales of Measurement (Steven, 1946)

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Fundamentals of Research in Education

​Characteristic of Scale	Nominal​ Scale	Ordinal Scale​	Interval Scale​	Ratio Scale​
​Applies names of numbers to categories?	Yes​	Yes​	Yes​	Yes​
Orders categories according to quantity?​	​No	​Yes	​Yes	Yes​
Displays equal intervals between consecutive numbers?​	No​	No​	​Yes	Yes​
​Displays a "true zero point"?	No​	No​	​No	Yes​
Examples​	​Gender Academic Major	​Ranking Single-item rating scale	Multiple-item summated rating scale Objectively scored tests	Physical measurements Count (number of occurrence of on event)​

Multiple Choice

Decide which scale of measurement the responses obtained from the following questionnaire item belongs to:

Do you plan to sit for MUET in the near future? Yes / No

• Nominal

• Ordinal

• Interval

• Ratio

Multiple Choice

Decide which scale of measurement the responses obtained from the following questionnaire item belongs to:

Number of times you have sat for MUET: ______

• Nominal

• Ordinal

• Interval

• Ratio

Multiple Choice

Decide which scale of measurement the responses obtained from the following questionnaire item belongs to:

Obtained Score for MUET Reading Test: ______

• Nominal

• Ordinal

• Interval

• Ratio

Multiple Choice

Decide which scale of measurement the responses obtained from the following questionnaire item belongs to:

MUET Band: 1 / 2 / 3 / 4 / 5 / 6

• Nominal

• Ordinal

• Interval

• Ratio

Some definitions

• ​Parameter = a value that describes some characteristic of a population, e.g. μ

• ​Statistic = a value that describes some characteristic of a sample, e.g. M

• Descriptive statistics = a value that is computed by analysing sample data and is used merely to describe some characteristics of that sample (without making any inferences about the larger population from which the sample was drawn)

• Measures of central tendency = statistics that reflect the most typical value in a distribution

• Measures of variability = statistics that indicate the amount of spread / dispersion that is displayed by a distribution of scores

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Fundamentals of Research in Education

Measures of central tendency

• Mean ( Symbol: M or x̅ )

  • summing the scores and dividing by the number of scores: x̅ = ∑x ∕ N ​

  • appropriate when:

    • it is an interval scale or ratio scale variable

    • the distribution of scores is approximately normal

• Median ( Symbol: Mdn or P₅₀ or Q₂ )

  • the score at or below which 50% of the scores in a distribution fall​

  • appropriate when:

    • it is an ordinal scale variable

    • the distribution is skewed​

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Fundamentals of Research in Education

Measures of central tendency

• Mode ( Symbol: Mo )

  • the most frequently-occurring value in a distribution

  • one mode = unimodal

  • two modes = bimodal

  • appropriate when:

    • it is a nominal scale variable​

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Fundamentals of Research in Education

Measures of central tendency

Task​

Using the ​MUET results dataset,

1. Clean the data

2. Determine the most appropriate measures of central tendency for each of the variables

3. Compute the relevant measures of central tendency for each of the variables

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Fundamentals of Research in Education

Measures of variability

• Range = stated as two numbers, i.e. the highest number & the lowest number / as one number, i.e. the difference between the highest & lowest numbers

• Variance = the average of the squared deviations from the mean

  • 3 different types of variance:

    • population variance = obtained when analysing all observations in a population: σ² = ∑(x−μ)² / N

    • sample variance = biased estimate of a population variance:

      S² = ∑(x−x̅)² / N

    • unbiased estimate of population variance: s² = ∑(x−x̅)² / (N−1)

• Standard deviation ( SD ) = square root of the variance (3 types : σ , S , s)

  • ​preferred because it is derived from the variance & easier to comprehend

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Fundamentals of Research in Education

Measures of variability

Task​

Using the ​MUET results dataset, assuming that the data were collected from a sample,

4. ​determine the variables where variance and standard deviation are appropriate

5. set up the table to calculate the variance for one of these variables

6. compute the sample variance and the corresponding standard deviation

7. compute the unbiased variance estimate​ and the corresponding standard deviation

8. compare the values obtained and explain when to use them

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Fundamentals of Research in Education

z score

• raw score = a participant's original score on some variable

• z score = ​a transformed version of the raw score indicating the number of standard deviations that the raw score falls above (or below) the mean

• z_x = (x−x̅) / S

• z score variable always has M = 0, SD = 1

• standardizing the variable = the process of converting all of the scores in a sample into z scores

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Multiple Choice

For z score calculation, which standard deviation do we used?

Clue: z_x = (x−x̅) / S

• population standard deviation

• sample standard deviation

• unbiased standard deviation estimate

z score

Task​

Using the ​MUET results dataset,

9. ​standardize all the MUET scores by converting them into z scores

10. ​calculate the mean and the standard deviation of the z score variables

11. Look at the different MUET scores. Is it easier to compare the MUET performance using raw scores or using z scores?

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z score & Standard normal distribution

• standard normal distribution = a theoretical perfect normal distribution in which all of the scores are z scores

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​What are the properties of the standard normal distribution that you can see from the diagram?

z score & Standard normal distribution

• Properties of the standard normal distribution

  • bell-shaped

  • symmetrical

  • unimodal

  • mean = median = mode

  • asymptotic = the curve stretches outward and downward as it moves away from the mean into the tail, but it never quite becomes a straight line and never actually touches the horizontal axis below it

  • standardised so that all the scores are z scores, i.e. μ = 0, ​σ = 1

  • area under the curve = percentage of distribution captured by specific z scores

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Fundamentals of Research in Education

z score & Standard normal distribution

• ​Important z scores especially in performing significance tests & creating CI

  • 95% of scores will fall in the interval from z = -1.96 to z = +1.96

  • 99% of scores will fall in the interval from z = -2.58 to z = +2.58

  • 99.9% of scores will fall in the interval from z =-3.29 to z = +3.29​​

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Fundamentals of Research in Education

More definitions

• Inferential statistics (aka parameter estimates / estimators) = a value that is:

  • ​computed by analysing sample data, but is the used to

  • estimate some likely characteristic of a larger population

• Statistical significance = If the results are statistically significant, the results observed in the sample would be unlikely if the null hypothesis were true.

• Research hypothesis = an educated guess regarding the likely answer to RQ

• Statistical null hypothesis (H₀) = a statement of "no effect", "no difference", or "no relationship"​ that the researcher typically hopes to reject

• Statistical alternative hypothesis (H₁) = ​a statement that there IS "an effect", "a difference", or "a relationship" that is usually consistent with the research hypothesis (and the researcher hopes to support)

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How to interpret statistical significance

1. ​Determine the significance level aka the alpha level, α, i.e. the size of the region of rejection in a sampling distribution. Usually α=.05 or α=.01

2. Decide if it is a two-tailed test (nondirectional) or a one-tailed test (directional).​

3. (a) Compare the​ obtained statistic against a critical value of the statistic found in the table of critical values

   OR

   (b) Consult the p value, i.e. the probability of obtaining a statistic the size of the current obtained statistic if the null hypothesis were true

4. Reject the null hypothesis ​(a) the obtained statistic > the critical value

   OR (b) p < α ​

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More definitions

• Test of differences = a statistical procedure used if:

  • your data are divided into two or more conditions

  • you plan to compute some measure of central tendency for each condition, and

  • you wish to determine whether there is a significant difference between these conditions with respect to the measure of central tendency

  • answers RQ like "Is there a difference between the means displayed by Condition 1 versus Condition 2?"

  • Parametric test: t-test

  • Nonparametric test: Kruskal-Wallis test

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Fundamentals of Research in Education

More definitions

• Parametric tests = data analysis procedures that have fairly strict assumptions involving population parameters:

  1. Interval-scale or ratio-scale measurement

  2. ​Normally distributed data

  3. Independence of observations

  4. Homogeneity of variance =

     • sample data in various treatment conditions must come from populations with equal variances (in tests of differences)

     • the variance of one variable should be the same at each level of the other variable (in tests of association) = homoscedasticity

• Nonparametric tests = do not require the above assumptions to be met

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Fundamentals of Research in Education

t-test

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Kruskal-Wallis test

• ​One variable with two or more ordered levels AND one nominal variable (2 or more groups)

• Determine whether the medians​ of two or more groups are different

• Calculate the H statistic

• ​H₀ : Mdn₁ = Mdn₂

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Fundamentals of Research in Education

More definitions

• Test of association = a statistical procedure used if:

  • you are analyzing just one group of participants, and

  • you wish to determine whether there is a relationship between two variables within that group

  • answers RQ like "Is there a relationship between Variable 1 and Variable 2?"

  • Parametric test: correlation

  • Nonparametric test: chi-square

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Fundamentals of Research in Education

Correlation coefficient

• Several types, most popular:

  • Interval/ratio scale: Pearson product moment correlation

  • Ordinal scale: Spearman

• ​-1 < r < +1

• ​H₀ : ρ_xy = 0

• Example of guidelines​

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Fundamentals of Research in Education

• ​To determine how strong a relationship is between two variables

Chi-square test for independence

• To determine whether there is an association between nominal variables

• ​Utilizes a contingency table (aka crosstabulation) = an arrangement in which data are classfied according to two categorical variables

• ​H₀ : Variable A is independent of Variable B / Variable A is not associated with Variable B

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Fundamentals of Research in Education

Chi-square test for independence

• To determine whether there is an association between nominal variables

• ​Utilizes a contingency table (aka crosstabulation) = an arrangement in which data are classfied according to two categorical variables

• ​H₀ : Variable A is independent of Variable B / Variable A is not associated with Variable B

TSLB3143

Fundamentals of Research in Education

Reporting Statistics in APA Style | Guidelines & Examples

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