Qual and Quant Research Methods - Statistics

Question 1

Why are statistics necessary in political science research?

Answer 1

Stats are used to conduct quantitative analysis and understand raw data collected in research.

Descriptive stats are often used to describe the data collected even if the project uses mixed qual and quant methods.

Recent trends have increased the prevalence of statistical analysis in political science research in tandem with an increase in data availability.

Question 2

Is data usually in data matrix form in our research?

Answer 2

Yes

Question 3

Describe a data matrix… what are the rows, columns and cells?

Answer 3

Rows = observations (individuals, countries, elections…)

Columns = variables (income, age, level of education, campaign spending…)

Cells = represent the value of a variable for a specific observation (e.g. a specific individual’s income, a country’s GDP per capita…)

The specific format of the file depends on the program, it was created with (e.g. excel spreadsheet, state file…)

Question 4

What is nominal data?

Answer 4

Data where response categories cannot be placed in a specific order (you cannot judge the distance between categories) - they are just things

E.g. country of birth, ethnicity…

Question 5

What is ordinal data?

Answer 5

Data where response categories can be placed in rank order (but distance between categories cannot be measured mathematically - if lots of categories we sometimes treat them as continuous for analysis purposes)

E.g. linkert scale, rank preference, levels of education…

Question 6

What is quantitative (interval and ratio) data?

Answer 6

Responses are measured on a continuous scale with rank order - assuming uniform distance/interval between responses. Treated as continuous.

E.g. age in years, temperatures in degrees, 1-10 ranking, income in GBP…

Question 7

What are the three measures of central tendency?

Answer 7

Mean, median and mode

Question 8

What are the four measures of spread and position?

Answer 8

Range, standard deviation, percentiles and interquartile range

Question 9

What is the mean? How do you calculate it?

Answer 9

The simple average - you take the sum of all values (∑) in a sample, and then divide them by the number of observations (n)

Mean is denoted by 𝑥̅

The mean is only appropriate for quantitative - interval (or ratio)

Question 10

What is the median?

Answer 10

The observation in the middle when we rank/order all observations from lowest to highest (e.g. ages lowest to highest)

BUT if we have an even number of observations, take the mid-point between the two middle values

Appropriate for both interval and ordinal variables, but not nominal variables!

Question 11

Ordinal example: Imagine the same sample of 10 respondents and what social class they identify as (between working, middle, and upper) - can you take the mean and/or median?

Answer 11

No - because there are no numerical values to be summed…

However, you can take the median if we arrange the values in order

Question 12

What is the mode?

Answer 12

The mode is the value that occurs most frequently

If there are values that occur equally and more than the other values, this is called bimodal distribution

Appropriate for interval, ordinal AND nominal variables

Question 13

Nominal example: Imagine the same sample of 10 respondents and what region they live in - can you take the mean, median and/or mode?

Answer 13

Mean = no (there are no numeric values to be summed)

Median = no (there is no meaningful order to put the categories into)

Mode = YES (it is the response that occurs the most)

Question 14

How do you compute the new mean if the origin of measurement is shifted?

Answer 14

Say if next year all respondents are 1 year older and we want the new mean age the mean age will be 1 year greater

Also applies to the median and mode when used for interval (numeric) variables

Question 15

How do you compute the new mean when there is a change in scale?

Answer 15

Say we want to measure age in months rather than years, we can just multiply everyone’s age by 12 to get each respondent’s age in months

This also applies to the median and mode when used for interval (numeric) data

Question 16

How do you get the mean of two related variables?

Answer 16

To get mean of the sum, add the two means together

E.g. imagine variable age is actually composed of two variables: years spent in school and years not spent in school - once you get the means of the two variables separately you add them to get the mean of the sum of two variables

Does not work for mode and median

Question 17

Why/when may the mean be not as informative as the median?

Answer 17

Where there are strong outliers that may affect the sample…

The mean is often heavily influenced by outliers (observations that have extreme values), and where there are strong outliers, the median might be a better measure of central tendency, or of a ‘typical observation’

Question 18

Why/when may the median be uninformative?

Answer 18

If there are relatively few values and/or a lot of zeroes!

Here the mean is often far more informative.

Question 19

How do you calculate the range?

Answer 19

Largest value MINUS smallest value of a data set

Question 20

Can two samples have the same mean but different ranges?

Answer 20

Yes

Question 21

Why/when may the range be uninformative?

Answer 21

The range is extremely sensitive to outliers - the range may not represent the spread of the majority of the data

E.g. could have a data set of 1, 2, 2, 3, 5, 6, 6, 29 - range would be uninformative

Question 22

What do percentiles divide the data set into?

Answer 22

Distributions of 100ths

First percentile of the data is the first 1%, second percentile is 2%, and so on… median percentile is the 50th percentile

Question 23

What are quartiles in data sets?

Answer 23

Divides the data into quarters

The inter-quartile range is oftentimes more information than the range and often presented as a box-plot

Question 24

What is the ‘variance’ in data sets?

Answer 24

Variance is a measure of dispersion - it is a measure of how far a set of numbers is spread out from their average/mean value

You take the difference between each value and the mean (e.g. difference between age 19 and mean of 28 is -9)… you then square each of the differences making them all positive (prevents the sum from being zero)…

Then we sum the difference and divide by n-1 for a sample of a population (and by n if we have the entire population)…

If the sum of distances from the mean squared is 656 you divide it by the sample number of 10-1, so 656 divided by 9 and you get a variance of 72.9

Question 25

What is standard deviation?

Answer 25

The square root of the variance - represents the typical distance of an observation from the mean As we calculated the variances using a square of differences, if we take the square root of the variance, then we get a measure of the rough typical distance of an observation from the mean So the standard deviation of our sample of 10 where the variance is 72.9, the standard deviation is 8.54, and so if our observations are ages we can deduce from the standard deviation that observations are typically around 8.54 years away from the mean (in either direction ofc)

Question 26

For what type of data is frequency and frequency distribution analysis useful for?

Answer 26

Categorical data It looks at how many observations are in each category - often looks at relative frequencies like the proportions or percentages in different categories

Question 27

How can relative frequency distribution be well presented?

Answer 27

With bar charts The height of the bar shows the frequency or relative frequency in that category - bars are also separate as to emphasise the it is a categorical variable E.g. each bar may represent the relative frequency of each social class respondents identify themselves as

Question 28

When are proportions particularly helpful?

Answer 28

When we have a dichotomous/dummy/binary variable (nominal variable with only 2 categories - e.g. yes/no, true/false...) Let's say we take the values of this variable 0 (no) and 1 (yes) and add up all 10 given values and divide by 10 (no. of respondents). If we have four 1s and 6 0s we find that the proportion of respondents that answered yes is 0.4.

Question 29

What are histograms?

Answer 29

A useful way to represent frequency distributions for quantitative variables Values of the variable on the x (horizontal) axis and how often each value occurs on the y (vertical axis) - useful for displaying frequency distribution of age, or maybe GDP per capita by country... (as the sample size increases, the sample distribution looks more like the population distribution and for continuous variables, if we imagine the sample size growing indefinitely, the shape of the histogram approaches a smooth curve)

Question 30

What is statistical inference?

Answer 30

The process of using what we know about a sample to make probabilistic statements about the broader population! Using probability theory, we can estimate what is going on in the population based on the sample

Question 31

What does statistical inference rely on?

Answer 31

Probability - the mathematical likelihood of a given event occurring (e.g. a proportion of a population voting for Donald Trump)

Question 32

What is a population parameter?

Answer 32

A quantity of the total population

Question 33

What is a sample statistic?

Answer 33

A quantity of the sample - this sample statistic provides an estimate of the population parameter

Question 34

What can probability theory be used for quantifying?

Answer 34

Quantifying the uncertainty that using a sample statistic brings

Question 35

What is probability in a random sample?

Answer 35

In a random sample or randomised experiment, the probability that an observation has a particular outcome is the proportion of times that outcome would occur in a very long sequence of similar observations. It is a number between 0 and 1 but in practice it is often expressed as a percentage.

Question 35

What is a probability distribution?

Answer 35

A probability distribution lists possible outcomes of an event and their probabilities (i.e. probability of each outcome) Assigns a probability to each possible value of a random variable - sum of the probabilities of each possible value equals 1! (i.e. 5 possible outcomes - can be 0.1, 0.2, 0.4, 0.1, 0.2)

Question 36

What is an example of a continuous variable?

Answer 36

Age? - theoretically can be an infinite number of ages

Question 37

What does a graph of the probability distribution of a continuous variable look like?

Answer 37

A smooth continuous curve (e.g. relative frequency distribution of age variable)

Question 38

Under the line of a continuous variable's probability distribution graph you may want to calculate an interval of variables (i.e. probability of ages 25-30), how do you calculate an interval?

Answer 38

The area under the curve for an interval of values represents the probability that the variable takes a value in that interval. The probability of the interval containing all the possible values equals 1 - so you add the probability of the individual variables to calculate the probability of the interval

Question 39

What is the empirical rule?

Answer 39

The empirical rule states that... approx 68% of values are within 1 standard deviation, 95% within 2 standard deviations, and 99.7% (almost all) within 3 standard deviations from the mean If a distribution is approximately bell-shaped on an x axis of 0-100 with a mean (y̅) of 50 and standard deviation (s) of 15 minus and 15 plus (35 to 65), then: About 68% of the observations fall between 35 and 65 ("y̅ − s" and "y̅ + s") About 95% of the observations fall between 20 and 80 ("y̅ − 2s" and "y̅+2s") All or nearly all observations fall between 5 and 95 ("y̅ − 3s" and "y̅ + 3s") Called the Empirical Rule because many frequency distributions seen in practice are approximately bell shaped. The standard deviation (s) is key, and so is the mean (y̅)! (e.g. exam results are on a bell curve)

Question 40

What is normal distribution in probability?

Answer 40

Normal distribution is the empirical rule applied to probability distribution It has the following characteristics: -Bell shaped curve -Distribution about the mean (mean, median, and mode are equal) -Defined by two parameters - the mean and standard deviation

Question 41

Why is normal distribution (normal probability distribution) important?

Answer 41

It approximates well sampled data in the real world AND we can use it to make statistical inference

Question 42

How can sampling distribution (and a wider sample set) help us to greater understand sample means?

Answer 42

The true mean may be 50 BUT WE DO NOT KNOW THIS... using random samples we have means of 47, 49, 51, 53... HOWEVER, we do not know how close these fall to the true population mean (50) - we can build a profile to better estimate the true value with the more samples we have! BUT using information about the spread of the sampling distribution we can predict how close it falls!

Question 43

What is central limit theorem in understanding sampling distribution?

Answer 43

As the number of samples increases, the sampling distribution approximates the normal distribution! This occurs even if the variable you are interested in is not nominally distributed - the mean of the sampling distribution is the population mean! More samples = more accurate reflection of true population mean (key assumption is that samples are randomly drawn from the population)

Question 44

What is a point estimate?

Answer 44

Point estimate - a single number that is the best guess for the parameter value

Question 45

What is a confidence interval?

Answer 45

Confidence interval - an interval of numbers around the point estimate that we believe (with some confidence) contains the parameter value Our confidence interval has a confidence level of any number (usually 90 something %) and this is the percentage of samples in which during repeat sampling, the parameter value falls within the interval

Question 46

How do we calculate a confidence interval?

Answer 46

Confidence interval... Lower point = point estimate - margin of error Upper point = point estimate + margin of error Point estimate ± Margin of error = ȳ ± t*(se)= ȳ ± t*𝒔/√𝒏

Question 47

What is the algebraic representation of the point estimate? (i.e. which letter)

Answer 47

ȳ

Question 48

How do you calculate the margin of error?

Answer 48

t*𝒔/√𝒏 Confidence level times by the standard error (standard deviation / square root of number of cases) t = assigned confidence level s = standard deviation √n = square root of number of cases

Question 49

What is a significance test?

Answer 49

A statistical significance test uses data to summarise the evidence about a hypothesis – by comparing point estimates of the parameters with the values predicted by the hypothesis! Involves... 1 - assumptions 2 - hypotheses 3 - test statistic 4 - p-value 5 - conclusion of statistical significance

Question 50

What kind of assumptions do we make in statistical significance tests?

Answer 50

-Type of data (quant or qual) -Randomisation (assumed random sampling maybe) -Population distribution (some tests assume a certain distribution) -Sample size (approx normal sampling distribution - but if sample large enough there is no need for normal population distribution)

Question 51

What is the null hypothesis and the alternative hypothesis?

Answer 51

Null hypothesis 𝐻𝑜: a statement that the parameter takes a particular value (e.g. the difference between men and women's salaries is 0) Alternative hypothesis 𝐻𝑎: the parameter falls in some alternative range of values (e.g. the difference between men and women's salaries is higher/lower than 0 - it is not 0)

Question 52

What does a significance test do in regards to comparing and analysing the null vs alternative hypothesis?

Answer 52

A sig test analyses the sample evidence about the null hypothesis, and investigates if the data contradicts the null hypothesis which would suggest the alternative hypothesis to be true It provides proof by contradiction of the null hypothesis - null hypothesis presumed to be true. Under this presumption, if the data observed is very unusual, we reject the null.

Question 53

What does the test statistic do in analysing the estimate vs the parameter value?

Answer 53

The test statistic summarises the entire data set. The test statistic compares (1) how much variation in Y can be explained by X - the regression slope estimate - and (2) how accurately we can measure the population slope - the standard error It displays how consistant with, or how far the estimate falls from the parameter value in 𝐻𝑜 (null hypothesis) - it is the number of standard errors between the estimate and the null hypothesis and the 𝐻𝑜 value e.g. t = 1.22 or t = 7.15

Question 54

What is the p-value in relation to the test statistic?

Answer 54

Probability we would have obtained the sample if the null hypothesis were actually true The p-value is the probability that the test statistic equals the observed value, or a value even more extreme in the direction predicted by 𝐻𝑎 (alternative hypothesis) The smaller the p-value, the stronger the evidence against the null hypothesis - smaller p value means more confidence - i.e. 0.05 p value = 95% confidence and 0.01 = 99% A moderate to large p-value means the data is consistent with 𝐻𝑜

Question 55

What is the 𝛼 (a) level in regards to the p-value?

Answer 55

The a level is the boundary value of 0.05 - if the p-value is below/equal to this a-level of 0.05 (i.e. 95% confidence) we can reject the null hypothesis! (i.e. the test is of statistical significance) If the p-value is above this 0.05 a-level then there is not enough proof by contradiction to reject the null hypothesis The smaller the 𝛼−level the stronger the evidence must be to reject 𝐻0. To avoid bias in the decision-making process you select 𝛼 before analysing the data!

Question 56

Do we ever say that we accept the null hypothesis?

Answer 56

No - we say that we fail to reject it when the p-value is above the a-level! We cannot be sure that the null hypothesis is correct - our data just does not adequately reject it

Question 57

What are the three key components of statistical association we need to think about?

Answer 57

Nature/Direction - how do the variables actually affect each other (e.g. does a higher income increase how right wing you are) Strength of relationship - how strongly does variable A affect variable B Statistical significance - how likely is it that the association you observe in a sample generalises to the population

Question 58

What is the simplest way of describing (or showing) a linear relationship?

Answer 58

As a linear relationship - with a straight line

Question 59

What actually is a linear regression?

Answer 59

A linear regression gives us the best linear association between two variables

Question 60

What is the formula of a linear function?

Answer 60

𝒚=𝜶+ 𝜷𝒙 y = the dependent variable x = the independent variable a = intercept/constant (value of y when x = 0) B = slope of gradient (how much y changes when x increases by 1 - higher = stronger affect of x)

Question 61

How do we rewrite the linear function to show the EXPECTED value of the dependent variable (as real world data is messy, not deterministic)?

Answer 61

E(𝒚) = 𝜶 + 𝜷𝒙 E(𝒚) is the mean of y for a given x - we account for variation around the regression line Linear regression does not make exact predictions; it predicts an average value of Y for a given X

Question 62

What is a least squares estimation (or aka the sum of squared errors)?

Answer 62

The line that (under some assumption) best fits the data hand It is th mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets ("the residuals") of the points from the curve = ∑(𝑦 − 𝑦 ̂)^𝟐 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛 𝑒𝑟𝑟𝑜𝑟 (distance between an observed value and the predicted value) = 𝑦 − 𝑦 ̂

Question 63

What is the sign of the slope coefficient (𝜷)? What does it tell us if 𝜷 is > 0 OR < 0 OR =0

Answer 63

Tells us about the nature (direction) of linear associations 𝛽 > 0 = positive relationship (as X increases, so does Y) 𝛽 < 0 = negative relationship (as X increases, Y decreases) 𝛽 = 0 = independence (as X increases, Y stays the same) Size of the 𝜷 gets bigger with a stronger linear association Something to bear in mind is that the size of 𝜷 depends on measurement units

Question 64

What does Pearson's r summarise/represent?

Answer 64

Pearson's r summarises the association between two quantitative variables in a single number. It is calculated using a standardised regression slope on a range of -1 to +1 i.e. it is the correlation between x and y, denoted by r

Question 65

Pearson's r is on a scale of -1 to +1, if r < 0, r > 0, and r = 0, what does this mean regarding the association between the variables?

Answer 65

r = 1 = perfect positive correlation r = > 0 = positive association (over 0.40 is strong, 0.20 is moderate, below 0.20 is weak) r = 0 = no association r = < 0 = negative association r = -1 = perfect negative association

Question 66

How do you represent the null and alternative hypotheses with 𝜷 (sign of the slope coefficient)?

Answer 66

Null hypothesis: 𝜷 = 0 Alternative hypothesis: 𝜷 ≠ 0

Question 67

What does a coefficient plot ('dot and whisker') show when regression results are plotted?

Answer 67

A point is plotted - this can be seen as the point estimate, BUT with a confidence interval (i.e. upper and lower bound of margin of error are plotted around the main point) - it displays uncertainty to help put data in greater perspective

Question 68

What does statistical control do?

Answer 68

It measures and accounts for potential confounders (z) in observational research - i.e. does the association between X and Y remain after controlling for Z?

Question 69

How can statistical control be implemented?

Answer 69

Using a straightforward extension of a simple linear regression known as a multiple linear regression function

Question 70

What does a multiple linear regression function help us do?

Answer 70

It helps us to engage in statistical control and rule out confounders (z) Interpretation remains the same, there are just new predictor variables (k)