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Research questions:

Research questions and hypotheses

Some other examples [made up data]:

Hypothesis: The more we procrastinate, the more stressed we feel.

Research questions and hypotheses

Some other examples [made up data]:

Hypothesis: The more we procrastinate, the more stressed we feel.

Research questions and hypotheses

Some other examples [made up data]:

Research question: Is there a relationship between caffeine consumption and productivity?

Research questions and hypotheses

Some other examples [made up data]:

Research question: Is there a relationship between caffeine consumption and productivity?

Research questions and hypotheses

Analysing Data Roadmap

Quantitative research

• In quantitative research, we often (but not always):

  • Start with a theory,

  • Devise an experiment (or a cross-sectional study) to test that theory

  • Collect the data

  • Describe our sample <- last term

  • Test hypotheses <- this term

A study

What can we say about this sample?

• On average, how long do cats spend on a task before looking at their owner?

• What is the shortest and longest time?

• What is the variance of scores in our sample? How do scores in our sample differ from each other?

• Are there any “unusual” cats in our sample?

The mean, the median and the mode

The mean, the median and the mode

Variance

• Variance - how scores differ from each other \(\frac{\sum{(X_i - X)^2}}{n-1}\) . Alternatively:

var(cat_sample$look_latency)

[1] 22.17848

Unusual cases?

• Given our sample, how unusual is a cat who took longer than 25 seconds to look at their owner

nrow(cat_sample)

[1] 21

There are 21 cats in our sample

dplyr::filter(
  cat_sample, 
  look_latency > 25
)

2 cats out of 21 represents a proportion of 2/21 = 0.095.

The empirical probability that a cat takes more than 25s to look at owner is 0.095, or 9.5 percent.

Populations vs samples

Sample =/= Population

Two cornerstones of statistical research:

1. A distribution of a sample from a given population will resemble the shape of that population.
2. A lot of variables have population distributions with a predictable shape.

Populations vs samples

1. A distribution of a sample from a given population will resemble the shape of that population.

• The larger our sample, the closer it will resemble the population distribution.

Populations vs samples

1. A distribution of a sample from a given population will resemble the shape of that population.

• The larger our sample, the closer it will resemble the population distribution.

Known distributions

2. A lot of variables have population distributions with a predictable shape.

One shape to rule them all

Sneaky distributions

• There are infinite possible combinations of means and standard deviations

• Therefore there are infinite possible normal distributions

• But not every bell-shaped distributions is a normal distribution

• Proportions matter

How to tell whether something is actually normally distributed

• We know that normal distribution has:

  • More scores in the middle

  • Fewer and fewer scores in the tails, the further away we get from the mean - the centre of the distribution

Z-scores

A way of assessing how unusual/uncommon a score is with reference to the mean

• Mean of our sample: 16.56

• SD sample: 4.71

\[ Z = \frac{X-M}{SD} \]

We are converting scores into standard deviation units.

Z-scores - what’s “unusual”?

Z-scores - what’s “unusual”?

Calculate Dracula’s Z-score:

\[ Z = \frac{X-M}{SD} \]

\[ Z = \frac{7-16.56}{4.71} \]

\[ Z = -2.03 \]

Dracula’s looking latency is 2.03 standard deviations smaller than the mean.

Z-scores - what’s “unusual”?

We can convert the whole sample into Z-scores:

Standardisation - the shape of the distribution remains the same but mean and SD change.

• Mean = 0

• SD = 1

Z-scores - what’s “unusual”?

We can convert the whole sample into Z-scores:

Standardisation allows us to work with probabilities:

• ~68% of scores will be within Z-scores of -1 to +1

• ~95% of scores will be within Z-scores -1.96 to + 1.96

• ~99% of scores will be within Z-scores -2.58 to + 2.58

Z-scores - what’s “unusual”?

We can convert the whole sample into Z-scores:

Standardisation allows comparisons:

• Across measurement scales

• Accounting for the characteristics of a normal distribution

• With reference to the population

Unusual in a population…

• Problem: we don’t know what the population looks like - i.e. what is the mean and the SD?
• We can assume a standard normal population distribution
  • Simplest normal distribution - makes minimal assumptions:
    
  • \(\mu\) = 0 (population mean)
  • \(\sigma\) = 1 (population SD)
• A distribution of Z-scores from a sample can be (1) compared against a standard normal population distribution (2) calculate the probability of a score based on standard deviation expectations.

This allows us to quantify whether something “unusual” or “surprising” with reference to the population.

Is Dracula a strange cat?

1. Place Dracula’s score on a standard normal distribution
2. Workout the probability of the shaded area
3. Compare against some defined “unusualnes” cut-off

Is Dracula a strange cat?

1. Place Dracula’s score on a standard normal distribution

Is Dracula a strange cat? Working out probabilities

2. Workout the probability of the shaded area

Option 1: Use a “Z-table” - ancient tablets from the 1800 BC. Pre-calculated and printed in special books or at the end of textbooks.

Option 1: Z-tables

Option 1: Z-tables

\[ Z = \frac{X - Mean}{SD} = \frac{7 - 16.56}{4.71} = -2.04 \]

Is Dracula a strange cat?

Another example

• Oreo looked at his owner after 22.8 seconds. Is Oreo unusual - e.g. in the top 5% ? 👍 👎

Another example

• Oreo looked at his owner after 22.8 seconds. Is Oreo unusual - e.g. in the top 5% ? 👍 👎

The other way around …

qnorm(p = 0.05, mean = 16.6, sd = 4.7, lower.tail = FALSE)

[1] 24.33081

Assuming that…

We calculated these probabilities assuming a certain shape of the population distribution - we don’t know whether this assumption is reasonable.

Summary

• Samples resemble populations - larger samples resemble them better

• This allows us make assumptions about what the population looks might look like and calculate probabilities.

• We assume some version of the reality (the population) and then we check whether what we observed is sufficiently unusual/interesting in this version of the reality.

• Comparing against a pre-decided critical cut-off from some population distribution is a core principle in hypothesis testing - more in the weeks to come.