Ipres 3

Question 1

Descriptive statistics

Answer 1

is a set of methods used to describe data and their characteristics. For example, if you were investigating the number of visitors to a beach
in August (nice job if you can get it!), you might draw a graph to see how the
number of visitors varied each day, work out how many people visit on an average
day and calculate the proportion of visitors who were male/female or children/
adults. These would all be descriptive data.

Question 2

Inferential statistics

Answer 2

involves using what we know to make inferences (estimates
or predictions) about what we don’t know. For example, if we asked 200 people
who they were going to vote for on the day before a local election we could try to
predict which party would win the election. Or if we asked 50 injecting drug users
whether they share injecting equipment such as needles with other users, we could
try to estimate the proportion of all injecting drug users who share equipment.

Question 3

Continuous variables

Answer 3

include things like ‘weights of newborn babies’, ‘distance travelled to work by those in full-time employment’, or ‘percentage of children living in lone-parent families’. Such variables are measured in numbers, and an observation may take any value on a continuous scale. For example, distance trav- elled to work could take a value of 0 miles for people working at home, 1.6 miles,
4.8 miles, or any other value up to 100 miles or more for those commuting long
distances. Similarly, any variable measured as a percentage can take a value of 0%,
100% or anything in between. For continuous variables the standard rules of arithmetic apply, so it makes sense to say that if you commute for 4 miles then that is
twice the commute of someone who commutes 2 miles.

Question 4

Discrete/ categorical variables

Answer 4

are not measured on a continuous
numerical scale. Examples of discrete variables are:
- sex: female/male
- religion: Buddhist/Christian/Hindu/Jewish/Muslim/Sikh/other
- degree subject studied: Politics/Sociology/Social Work
and so on. Such variables have no numeric value. We may assign them a number,
for example, Politics = 1, Sociology = 2, Social Work = 3 and so on, but the actual
numbers do not mean anything. For example, Social Work is not three times greater
than Politics!

Question 5

types of categorical variables

Nominal variables

Answer 5

are variables that have two or more categories, but which do not have an intrinsic order or inherent numerical quality in themselves. So nominal variables include things like ‘marital status’, ‘ethnicity’ or ‘location’, in addition to the variables already identified including ‘religion’ and ‘degree studied’. In some cases there may be many attributes to a nominal variable. For instance, if we were classifying where people live in the USA by state there would be 50 attributes (or states).

Question 6

types of categorical variables

Dichotomous variables

Answer 6

are nominal variables which have only two categories or levels. For example, if we were looking at gender, we would generally categorise somebody as ‘male’ or ‘female’. This is also a nominal variable. A further example would be if we asked somebody if they had ever smoked, giving them the possible answers of ‘yes or ‘no.

Question 7

types of categorical variables

Ordinal variables

Answer 7

have two or more categories, like nominal variables, but the categories
can also be ordered or ranked moving from greater to smaller values (or vice versa). So an opinion poll might ask how likely you were to vote for a particular party at the next election with the possible options ‘very likely’, ‘likely’, ‘not sure’, ‘unlikely’ or ‘very unlikely’. While the responses are in a particular order we cannot (or should not!) place a ‘value’ on them. This is despite our own views about which we prefer! Another example would be if you asked someone to provide an answer to the statement ‘I generally eat healthily’ and had a similar scale from ‘strongly agree’ to ‘strongly disagree’. It is difficult to say whether the distance between the two categories is equal as it will depend upon individual perception. So someone who has their ‘five-a-day’ and five chocolate bars too may consider themselves to eat healthily, whereas another person may consider that eating two chocolate bars in addition to lots of fruit and vegetables means they do not eat healthily. As a result, the distance between the points on the scale is not clear and continuous.

Question 8

continupus variables

difference between interval and ratio variables

Answer 8

interval has 0 as a an arbitrary point (celcius scale) ,ratio does not (age, weight)

Question 9

Item non-response

Answer 9

A specific type of missing data that occurs when a respondent does not provide a valid answer to a question, which can reduce sample size and introduce bias.

Question 10

Selection bias

Answer 10

A type of bias that arises when missing data is not random, meaning that certain groups of people or opinions are systematically underrepresented.

Question 11

Listwise deletion

Answer 11

A method of handling missing data by removing all cases with missing values from the analysis, which can reduce sample size and potentially introduce bias.

Imagine you’re playing a board game with your friends, but one of them doesn’t answer an important question. Instead of guessing their answer, you decide to remove them from the game completely. That’s what listwise deletion does—it removes any data that has a missing answer, even if most of it is still useful.

Question 12

Imputation

Answer 12

A technique for handling missing data by predicting the missing responses using other available information from the questionnaire.

Let’s say you’re reading a storybook, but one page is missing. Instead of skipping that part, you try to guess what happened based on the rest of the story. That’s what imputation does—it fills in missing answers by using the information that’s already there.

Question 13

Mean imputation

Answer 13

A basic form of imputation that replaces missing values with the midpoint or mean, which is only reliable if missing data is random.

Question 14

Regression imputation

Answer 14

A more advanced method that uses regression techniques to estimate missing values based on other answers from the respondent, making it more statistically sound but requiring good predictors.

Question 15

Aggregate data

Answer 15

means collecting and combining lots of individual pieces of information to look at overall patterns instead of specific details.

Imagine a jar full of different-colored candies. Instead of counting each color one by one, you just say, “There are 100 candies in total, and about 40% are red.” That’s aggregation—you’re summarizing the data instead of focusing on each tiny piece.

Question 16

Likert scale (interval data)

Answer 16

A Likert scale is a rating scale used in surveys to measure people’s opinions, attitudes, or behaviors. It typically consists of a series of statements with response options ranging from strongly agree to strongly disagree (or similar). It helps quantify subjective feelings in a structured way.

Question 17

What is skewness?

Answer 17

Skewness is about how the data in a graph “leans” or is “stretched” to one side. Imagine stacking blocks from smallest to biggest, and then a few really big or really small blocks change the shape.

Question 18

Positively skewed (right-skewed)

Answer 18

• Most data is small and clumped on the left side of the graph.
• A few really big values stretch the graph to the right.
• The mean (average) is pulled up by the big numbers.
• The median (middle value) stays closer to the center of the data.

👉 Result:
Mean > Median

📌 Why?
Because the mean adds everything up and divides, those few very large values make the total bigger, so the average increases.

Example:
Imagine these numbers: 2, 3, 4, 5, 30

Mean = (2+3+4+5+30)/5 = 8.8

Median = 4
→ Mean is higher than median.

Question 19

Negatively skewed (left-skewed)

Answer 19

• Most data is large and clumped on the right side of the graph.
• A few really small values stretch the graph to the left.
• The mean gets pulled down by the small numbers.
• The median stays closer to the middle of the data.

👉 Result:
Mean < Median

📌 Why?
Because the mean is dragged down by those very small numbers, even if most values are higher.

Example:
Numbers: 1, 20, 21, 22, 23

Mean = (1+20+21+22+23)/5 = 17.4

Median = 21
→ Mean is lower than median.

Question 20

Guidelines for choosing a measure of the average

Answer 20

• Modes are not used very often, though they can be useful in certain circumstances. Avoid them ingeneral or use them along with other measures!
• The median is more intelligible to the general public because it is the ‘middle’ observation.
• The mean uses all the data, but the median does not. Therefore the mean is‘ influenced’ more by unusual or extreme data. If the data are particularly subject to error, use the median.
• The mean is more useful when the distribution is symmetrical or normal. The median is more useful when the distribution is positively or negatively skewed, because outliers do not affect the median.
• The mean is best for minimising sampling variability. If we take repeated samples from a population, each sample will give us a slightly different mean and median. However, the means will vary less than the medians. (For example, if we took 10 different samples of 20 students and calculated the average weight for each of the 10 groups, the mean weights would differ less between the 10 groups than the median weights.)

Question 21

other distributions of data

Answer 21

• deciles (which divide a distribution into ten equal parts)
• quintiles (which divide a distribution into five equal parts)
• quartiles (which divide a distribution into four equal parts)

Question 22

Percentiles

Answer 22

Percentiles tell you how a value compares to the rest of the data by showing the percentage of values that fall below it. If you’re in the 90th percentile on a test, that means 90% of the class scored lower than you — and only 10% scored higher.

Suppose we had a set of exam marks and we knew that the 30th percentile was a mark of 52. This means that 30% of people who sat the exam got a mark of 52 or lower.

Question 23

Range

Answer 23

The difference between the highest and lowest values in a dataset, used to measure the spread of data.

Question 24

Inter-Quartile Range (IQR)

Answer 24

The difference between the upper quartile (Q3) and lower quartile (Q1) in a dataset, representing the range of the middle 50% of the data.

To work it out we would position the variables in order from highest to lowest and concentrate on the middle 50 per cent of the distribution. So the inter-quartile range is the range of the middle half of the observations, the difference between the lower quartile and the upper quartile.

Question 25

Histogram vs. Density Plot

Answer 25

A **histogram** is a type of graph used for interval or ratio data. It works by grouping data into bins (ranges) and displaying these bins as bars. The height of each bar represents how many values fall into that particular range, giving a visual overview of the frequency distribution of the dataset. A **density plot**, on the other hand, is a smoothed version of a histogram that shows the proportion of values at each point of a continuous random variable. Instead of grouping data into bars, it uses a curve to estimate the probability distribution, which helps to visualize where values are most concentrated and makes it easier to compare multiple distributions.

Question 26

Summary Statistics

Answer 26

*Summary statistics* are used to describe patterns in data and make comparisons across datasets. They help simplify large amounts of information by highlighting key characteristics of a distribution. There are two main types: a. **Central Tendency** – Measures that describe where the center of a distribution lies, or what a "typical" value might be. This includes: - Mode: The most frequent value - Mean: The arithmetic average - Median: The middle value when data is ordered b. **Dispersion** – Measures that show how spread out or clustered the data values are, such as range, interquartile range (IQR), variance, and standard deviation.

Question 27

Properties of mean vs median (non-) resistance to outliers

Answer 27

* Median is resistant to outliers, mean is not. * Perfectly symmetric: mean = median * Skewed to the left / negatively skewed: mean < median * Skewed to the right / positively skewed: mean > median

Question 28

Why Use n - 1 in Standard Deviation

Answer 28

When calculating standard deviation for a sample, we divide by n - 1 instead of just n to correct for bias. This is called Bessel’s correction. It helps us get a more accurate estimate of the population standard deviation because using just n tends to underestimate the variability in the population. Subtracting 1 accounts for the fact that we're using the sample mean (an estimate) rather than the true population mean.

Question 29

Z score

Answer 29

* A Z-score measures the number of standard deviations an observation is away from the mean. * A positive Z-score shows that the observation isgreater than the mean (above average). * A negative Z-score shows that the observation is lower than the mean (below average). * The Z-score will be zero if the observation equals the mean. * Most Z-scores will lie in the range from Z = —2 to Z = 2. Values more than two standard deviations from the mean tend to be extreme values (outliers).

Question 30

Difference Between Standard Deviation and Z-Score

Answer 30

A standard deviation (SD) measures how spread out the values in a dataset are—it tells us how much the data tends to deviate from the mean. In contrast, a z-score measures how far a single data point is from the mean, expressed in units of standard deviation. For example, if someone has a z-score of 1.5, it means their value is 1.5 standard deviations above the average. While SD describes the variability of the entire dataset, the z-score focuses on the relative position of one specific value within that distribution.

Question 31

What does a Z-score of +1 mean. How much of the data lies between the mean and +1 SD on a normal distribution?

Answer 31

A Z-score of +1 means the value is one standard deviation above the mean. In a normal distribution (bell curve), approximately 34.1% of the data lies between the mean and one standard deviation above it. For example, if the mean is 70 and the SD is 10, then about 34.1% of values fall between 70 and 80.

Question 32

normal distribution characteristics

Answer 32

* The mean lies in the middle and the curve is symmetrical about the mean. The mean is equal to the median. * Most of the observations are close tothe mean, so the frequency ishigh around the mean. There are fewer observations that are much greater or much smaller than the mean. * The curve never actually touches the X axis on either side; itjust gets closer and closer.

Question 33

Types of errors in sampling

Answer 33

**Sampling Variability:** Refers to the natural differences in results (like the mean or standard deviation) that occur when multiple random samples are taken from the same population. These differences are due to chance and will vary depending on which individuals happen to be selected. **Sampling Error:** This is the difference between the sample estimate (like the sample mean) and the actual population value. It happens because a sample is only a part of the population, and it may not perfectly represent it. Sampling error can be minimized by using better sampling methods, but it is not caused by mistakes. **Non-Sampling Error:** These are errors unrelated to how the sample is selected, such as data entry mistakes, misunderstood questions, or biased data collection procedures.

Question 34

# probability sampling Simple Random Sampling

Answer 34

**Definition:** Every individual in the population has an equal chance of being selected.​ **Example:** Assign numbers to all 1,000 employees in a company and use a random number generator to select 100 employees for a survey.

Question 35

# probability sampling Systematic Sampling

Answer 35

**Definition:** Select every nth individual from a list, starting from a randomly chosen point.​ **Example:** In a list of 300 students, randomly choose a starting point between 1 and 15, then select every 15th student to form a sample of 20 students.

Question 36

# probability sampling Stratified Sampling

Answer 36

**Definition:** Divide the population into subgroups (strata) based on a specific characteristic, then randomly sample from each subgroup proportionally.​ **Example:** In a company with 800 female and 200 male employees, to create a representative sample of 100 employees, randomly select 80 women and 20 men.

Question 37

# probability sampling Cluster Sampling

Answer 37

**Definition:** Divide the population into clusters, randomly select some clusters, and then survey all individuals within those clusters.​ **Example:** To study university students across the UK, randomly select 15 universities (clusters) and survey all students within each selected university.

Question 38

# non-probability sampling convenience sampling

Answer 38

**Definition:** Sampling individuals who are easiest to access. No attempt is made to ensure representativeness. **Example:** A teacher surveys their own students to study young adult habits. 🧠 Pros: Quick and easy. Legitimate in pilot studies and experiments ⚠️ Cons: High risk of bias, not generalizable.

Question 39

# non-probability sampling snowball sample

Answer 39

**Definition:** Start with a few participants who meet your criteria, then ask them to refer others. **Example:** Interviewing marijuana users by first contacting known users, who then refer friends. 🧠 Pros: Useful for hidden/hard-to-reach groups (e.g., drug users, undocumented migrants). ⚠️ Cons: Not representative of the larger population.

Question 40

# non-probability sampling quota sample

Answer 40

**Definition:** Select participants non-randomly to fill predefined quotas (e.g., gender, age). **Example:** You want 60 men and 40 women in your sample of 100. You stop collecting data once the quotas are filled. 🧠 Pros: Ensures inclusion of specific groups. ⚠️ Cons: Still non-random, subject to interviewer bias.

Question 41

# non-probability sampling purposive sample

Answer 41

**Definition:** Select participants intentionally because they fit a specific profile relevant to the research. **Example:** Interviewing women aged 30–40 on the street for a study about skincare habits. 🧠 Pros: Good for targeted research. ⚠️ Cons: Subjective judgment, potential for selection error.

Question 42

central limit theory

Answer 42

The Central Limit Theorem (CLT) is a fundamental principle in statistics stating that, regardless of the population's distribution, the distribution of sample means will approximate a normal distribution as the sample size becomes large. This holds true even if the original data are not normally distributed. Typically, a sample size of 30 or more is considered sufficient for the CLT to apply. This theorem is crucial because it allows statisticians to make inferences about population parameters using the normal distribution, facilitating hypothesis testing and confidence interval estimation.​ Apply: - determine the mean using CLT, error of spmple - draw diagram - calculate the z-score (sample mean-population mean)/SE - look up probability in table, if needed deduct from the graph (0.5-Z) - draw conclusion

Question 43

SE

Answer 43

**standard deviation of the sample means**; works the same as SD in a normal distribution (68% of sample means lie within 1SE on each side)

Question 44

confidence intervals

Answer 44

A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter (such as a mean or proportion). It provides an estimated range believed to encompass the parameter with a certain level of confidence, commonly 95% or 99%.​ Wikipedia For example, a 95% confidence interval for a population mean might be 70.3 kg to 72.2 kg. This means that if we were to take many random samples and compute a confidence interval from each, approximately 95% of those intervals would contain the true population mean. ​ Wikipedia It's important to note that the confidence level (e.g., 95%) refers to the long-run success rate of the method used to construct the interval, not the probability that the true parameter lies within a specific interval calculated from a single sample.​ Confidence intervals are widely used in statistics to express the uncertainty associated with sample estimates and to infer population parameters.​ ## Footnote populatuon has to be normaly distributed ir have a sample>30

Question 45

on confidence intervals

Answer 45

**point estimate:**estimate of the mean (from the sample to population) **interval estimate:** the likley rangr in which we can be reasonably sure that the population will lie

Question 46

when to use SD,SE,CI

Answer 46

* Use SD when you want to describe the spread of data within your sample.​ * Use SE when you're interested in the precision of your sample mean as an estimate of the population mean.​ * Use CI when you want to express the uncertainty around your sample estimate and make inferences about the population parameter.​

Question 47

one tailed test

Answer 47

​A one-tailed test is used when the research hypothesis specifies a direction of the effect—either an increase or a decrease. Purpose: Tests for the possibility of an effect in one direction only. Null hypothesis (H₀): The parameter is equal to a specific value. Alternative hypothesis (H₁): The parameter is either greater than or less than that value, depending on the research question.​ Example: Testing if a new drug increases recovery rate compared to the standard treatment.​ Critical Region: Allotted entirely to one tail of the distribution.​ Consideration: One-tailed tests have more statistical power to detect an effect in one direction but cannot detect an effect in the opposite direction.​ ## Footnote Use a one-tailed test when the research hypothesis predicts a specific direction of the effect.

Question 48

two tailed test

Answer 48

A two-tailed test is appropriate when the research hypothesis does not specify a direction, meaning the effect could be in either direction. In a two-sided test, the aim is to discover whether the sample mean or proportion is different from the population mean or proportion, not whether itiseither larger or smaller. Purpose: Tests for the possibility of an effect in both directions.​ Hypotheses: Null hypothesis (H₀): The parameter is equal to a specific value. Alternative hypothesis (H₁): The parameter is not equal to that value.​ Example: Testing if a new teaching method leads to a different test score average, without specifying if it's higher or lower.​ Critical Region: Split between both tails of the distribution.​ Consideration: Two-tailed tests are more conservative and are appropriate when deviations in both directions are of interest.​ ## Footnote Use a two-tailed test when the research hypothesis does not predict a direction, or when deviations in both directions are important.​

Question 49

Cohen's D

Answer 49

is a statistical measure used to **quantify the effect size between two groups**. It expresses the difference between two means in terms of standard deviation units, allowing you to assess how large or meaningful that difference is, regardless of sample size. 0.2 = small effect 0.5 = medium effect 0.8 = large effect ## Footnote the more variation (SD) the smaller the effect

Question 50

tyoes of errors

Answer 50

🔴 **Type I Error (False Positive)** Definition: Rejecting the null hypothesis when it is actually true. Meaning: You conclude there is an effect or difference when there really isn’t. Probability: Denoted by α (alpha), usually set at 0.05 (5%). Example: A medical test says a person has a disease when they don’t. 🔵 **Type II Error (False Negative)** Definition: Failing to reject the null hypothesis when it is actually false. Meaning: You miss a real effect or difference. Probability: Denoted by β (beta). Power of a test: 1−β 1−β; the probability of correctly detecting a true effect. Example: A medical test fails to detect a disease that a person actually has. 1. **α** is concerned with making sure that the negative finding (meaning: no difference) we have is indeed TRUE ( and not a FALSE positive). But what about cases where we fail to detect an effect in our sample? Is that effect not there (in the population) or do we have too few observations to detect the effect (=false negative). This is where β comes in. 2. **β** is the probability of finding a false negative (a difference between groups that we fail to find). Power (1- β) describes the true positive probability- usually is 80%: 8 out of 10 times we are likely to pick up an effect, that in fact really exists (true positive). And 2 out of 10 times we fail to pick up an effect that in fact does exist (false negative) ## Footnote The goal of power calculation is to determine mostly how much observations our design should have to pick up a true positive

Question 51

chi square

Answer 51

The chi-square (χ²) value is a statistical measure used to assess the difference between observed and expected frequencies in categorical data. It is commonly applied in tests like the **chi-square test of independence, which evaluates whether two categorical variables are related, and the goodness-of-fit test, which checks how well an observed distribution fits a theoretical one**. In the formula O represents the observed frequencies and E the expected frequencies under the null hypothesis. A larger chi-square value indicates a greater discrepancy between observed and expected outcomes. This value is then compared to a critical value from the chi-square distribution table (based on the degrees of freedom and chosen significance level) to determine if the result is statistically significant and whether to reject the null hypothesis.

Question 52

how to use it

Answer 52

1.**Chi-Square Test of Independence (for contingency tables)** Null Hypothesis (H₀): There is no association between the two categorical variables. Or: The two variables are independent of each other. Example: If you're testing whether gender and voting preference are related: H₀: Gender and voting preference are independent. 2.**Chi-Square Goodness-of-Fit Test**: Null Hypothesis (H₀): The observed distribution matches the expected distribution. Or: The data follow the specified theoretical distribution. Example: If you're testing whether a die is fair: H₀: Each side of the die has an equal probability of occurring (i.e., 1/6).

Question 53

when to use it

Answer 53

Use this when you’re dealing with categorical variables — like gender (e.g., male/female/other) and voting preference (e.g., Party A/Party B/None). These variables do not have a numerical or continuous scale, so you can’t compute a correlation coefficient like Pearson’s r. ✅ Chi-square tells you if there’s a statistical association (i.e., dependence or independence) between two categorical variables.

Question 54

Mann Whitney U test

Answer 54

The Mann-Whitney U test is a non-parametric alternative to the independent samples t-test, used to compare whether two independent groups differ significantly in their central tendency when the data are not normally distributed or are ordinal. Instead of comparing means, it ranks all the values from both groups together and evaluates whether one group tends to have higher or lower ranks than the other. It is especially useful when the assumptions of normality and equal variances required for parametric tests are not met, making it a robust choice for skewed or ranked data.

Question 55

Wilcoxon matched-pairs signed-rank test

Answer 55

is a non-parametric test used to compare two related samples, such as measurements taken before and after an intervention on the same subjects. It is the non-parametric equivalent of the paired samples t-test and is used when the differences between paired values do not follow a normal distribution. The test works by ranking the absolute differences between paired observations, assigning signs based on the direction of change, and then analyzing these signed ranks to determine whether the median difference is significantly different from zero. It's especially useful for small samples or data measured on an ordinal scale. * The Wilcoxon signed-rank test tells you if the differences between two related groups (before and after) are significantly different from zero. * You calculate the sum of positive and negative ranks, and the smaller of those two sums is your test statistic (W). * You compare W to the critical value from a table. If it's smaller than or equal to the critical value, there's a significant difference.

Question 56

when to use each test

Answer 56

**T-Tests:** Use when you want to compare averages (e.g., two groups of people or before vs. after). **Mann-Whitney U:** Use when you have two groups but your data isn’t normal. **Wilcoxon Signed-Rank:** Use when comparing before/after data but your data isn’t normal. **Pearson/Spearman Correlation:** Use to see if two things (like height and weight) are related. **Z-Test:** Use when you have a large sample or know the population information, and you want to compare it to a sample. **Normal Distribution:** Most things follow a bell curve, like people’s heights. **Confidence Interval:** Tells you the range of answers where you’re fairly sure the true answer lies. **Test Statistic:** A way to measure how far your data is from what you expect. **Z-Test for Proportions:** Use when comparing percentages or proportions (like how many people prefer pizza).

Question 57

when do we use a box plot

Answer 57

when we want to present the median and inter-quartile range of a distribution

Question 58

about distributions

Answer 58

ND: mean=median left-skewd D: mean median uniform distribution: mean=median ! the median is resistant to outliers unlike the mean (so use the median when the data is skewd?)

Question 59

t-test

Answer 59

verifies wether there is a significant difference between the means of two groups-use 1.96 as Z value t value= difference between the means/ SE

Question 60

types of tests

Answer 60

parametric test: used when the data is normally distributed non-parametric: used when its not

Question 61

degrees of freedom

Answer 61

Degrees of freedom (often written as df) represent the number of values in a statistical calculation that are free to vary. When you calculate something like a mean, one value is always “locked in” once the others are known, because the total has to add up to a certain number. For example, if you know the average of three test scores must be 70, and you already chose two of the scores, the third score isn’t free to vary—it must be whatever value makes the average come out to 70. So in that case, you have 3 values but only 2 are free to vary, giving you 2 degrees of freedom. In general, for many tests like the t-test, the degrees of freedom equal the number of data points minus one (n − 1). Degrees of freedom are important because they help determine which values to use when looking up critical values in statistical tables and affect the shape of the distribution used in your test.