Introduction and Basics- Lecture 1 Flashcards
3 main roles in statistics
- Designing experiments
- Analysing data
- Drawing conclusions (understanding results)
Definitions:
1. Data
2. Statistics
3. Population
4. Sample
- consists of information that comes from observations, measurements, responses
- science of collecting, anlaysising and organising data. Involves interpretation
- collection of all outcomes, responses, measurements that are of interest
- subset of a population
Descriptive statistics
Involves organisation, summarization and display of data: e.g words, graphs, captions, numbers
Inferential statistics
Using a sample to interpret the results and draw conclusions about population
Design of a statistical study
- Identify variable of interest and population of the study
- Detailed plan to collect data- ensure sample is representative of population if using a sample
- Collect data
- Describe data
- Interpret data and make decisions about population using inferential statistics- drawing conclusions
- Identify possible errors
Methods of data collection - Observational study?
Researcher observes and measures characteristics of interest of part of population
Methods of data collection - experiment?
Treatment is applied to part of a population, and responses are observed
Methods of data collection - simulation?
use of a mathematical or physical model to reproduce the conditions of a situation or process
Methods of data collection - survey
investigation of one or more characteristics of a population
1. census = measurement of an entire population
2. sampling = measurement of part of population
Defintion - stratified sample?
members from each segment of a population, to ensure each segment is represented
Defintion - cluster samples?
all members from randomly selected segments of a population
Defintion - systematic samples
each member of the population is assigned a number. Starting number is randomly selected and sample members are selected at regular intervals
Defintion - convenience samples?
only of availbale members of the population (can be used as a pilot study but it is not representative of the whole population, will be biased)
Discrete variable
indivisible categories e.g class size, number of children in a family
Continuous variable
infinitely divisible into whatever units e.g time, weight. Time can be measured to the nearest minute, second, half - second etc.
Measuring variables
Requires a set of categories = scale of measurement and a process that classifies each individual into one cateogry
4 Types of Measurement scales
1. Nominal scale
2. Ordinal scale
3 Inverval scale
4. Ratio scale
- unordered set of categories indentified only by name
- ordered set of categories
- ordered series of equal-sized categories
- interval scale where a value of zero indicates none of the variables
correlational study
determine whether theres a relationship between 2 variables, describe relationship and observe 2 variables as they exist naturally
manipulated variable
independent variable
observed variable
dependent variable
central value characterisation of whole set of data
measures of central value e.g mean or media must be coupled with measures of data dispersion (average distance from the mean) to indicate how well the central value characterises data as a whole. The smaller the narrow window data, the better the representation of data.
center measurement defintion
summary measure of overall level of dataset e.g mode, mean, median
median sensitivity?
median is less sensitive to outliers (extreme scores) than the mean, thus better measure than the mean for highly skewed distributions
variability (dispersion) measures what?
amount of scatter in a dataset with methods used to represent this: range, variance, IQR, coefficient of variation. Most common is standard deviation
what is variance?
variance of set of observations is the average of the squares of the deviations of the observations from their mean
standard deviation
square root of the variance and variance showing how the data varies across collection of sample set. Large standard deviation indicates data points are far from the mean
data collection and selection of sample sizes makes difference why?
if there are 9 samples, can be assumed as 1 dataset with N=9
BUT can also be assumed as 3 datasets from 3 independent studies, N=3
mean remains the same but changes standard deviation
standard error
standard deviation of sample means and a measure of how representative a sample is likely to be of the population
large standard error?
a lot of variability between the means of different samples, thus sample might not be representative of population
small standard error?
most sample means are similar to the population mean, thus sample is accurate reflection of population
frequency distribution- best visualisation of data?
Histogram, but number of bins are important. Histograms not good when you dont have enough data. (too many bins = noisy, too few bins can mask out important features)
1.normal distribution, 2.skewed distribution, 3.modality distribution
- central bellcurve shape uniform
- shifted to left (positive) or right (negative)
- 2 efective central values and 2 populations of responses
z scores?
used to convert any normal distribution such that:
- mean = 0
- standard deviation - 1
important z score: +- 1.96 (removes outlying data- 2.5%)
z score calculation?
𝑧=(𝑋−𝑋̅)/𝑠
null hypothesis?
nothing is happening
alternate hypothesis?
what you’re expecting to happen is happening, trying to disprove null hypothesis
p- value?
Probability that the observed statistic is equal to or more extreme, than observed result then Ho is true.
trying to find at which point you have enough evidence against null hypothesis to support actual alternate hypothesis.
smaller p value?
swinging against null hypothesis, further towards end of bell curve, stronger evidence against null hypothesis
one sided test
in particular condition experiment is set up with, only going one way
two sided test
not null hypothesis, can go up or down
for one sided test, critical value that represents z score, p value?
either + or - but not both
(p<0.5) if z>- 1,645)
for two sided test, critical value?
number that separates blue zone (ends of bell curve) from the middle. To be statistically significant, z score needs to be in the blue curve
p value larger than 0.05?
large p value, not disproving null hypothesis. Not good indication is that is it 2 significant means.
for 2 sided test, where p value is P= 0.037, what do you do and is there is strong evidence for or against null hypothesis, Ho?
If 2 sided, P= 0.037 x 2 = 0.07
Does not swing either way, thus not surveying enough people or greater number of population. With this p-value, cannot say we have enough evidence against null hypothesis. Right on borderline and no strong evidence in either direction
difference between p - value and a (alpha)-level?
a- level = indication error and is set before collection of data, to help set up experiment.
defines error we are willing to make to say we made a difference. If we’re wrong, its an alpha error
p- value = calculated after we gather data
Calculated probability of a mistake by saying it works e.g level of significance.
Descrives percent of population/ area under the curve in the tail that is beyond our statistic
a- level is 0.5. Reject Ho when?
P ≤ a so p is smaller than alpha
a- level is 0.5. Retain Ho when?
P>a
if a -level is small and tight data set, harder to reject null hypothesis
B (beta level)?
probability of erroneously retaining Ho
Type I error?
erroneous rejection of true Ho
Type II error?
erroneous retention of false Ho
Power?
1- B (beta)
probability of avoiding a type II error (retaining a false null hypothesis
1- B = Pr (reject Ho i I Hfalse)
True or False- all variables can be classified as quanititative or categorical value?
True
True or False- Categorical values can be continuous variables?
False
True or False- quantitative variables can be discrete variables
True
inferential statistics
used to make a conclusion about a population based on a sample dataset
descriptive statistics
involves the organisation, summarization, and display of data
center measurement
summary measure of the overall level of a dataset (mean, median, mode, geometric mean)
what is the better measure, mean or median
the median is less sensitive to outliers (extreme scores) than the mean and thus a better measure than the mean for highly skewed distributions
variability
measures the amount of scatter in a dataset
range
crude measure of variability
what does a large standard deviation signify?
the data points are far from the mean
p-value in context of null hypothesis
used to quanitfy the idea of statistical significance of evidence
why particular p-value
probability that results gained by chance and chance is the only factor
single sample
one group; no concurrent control group
paired sample
two samples; data points uniquely matched
two independent variables
two samples, separate (unrelated) groups
Measure vitamin content in loaves of bread and see if the average meets national standards
single sample as just one experiment done and one population
Compare vitamin content of loaves immediately after baking versus content in same loaves 3 days later
paired sample as comparison and 2 pairs
Compare vitamin content of bread immediately after baking versus loaves that have been on shelf for 3 days
independent as same thing not measured twice. Two groups with different reatments
degrees of freedom
number of observations in the data that are free to vary when estimating statistical parameters
df conservative
the smaller of (n1 – 1) or (n2 – 1)
comparison of means using t statistic
(𝑥̄1−𝑥̄2)±(𝑡(𝑑𝑓,1−𝛼/2))(𝑆𝐸(𝑥̄_1−𝑥̄_2 ))
difference between µ and X ̅
µ is the population mean and X ̅ is the sample mean
α-level represent?
The probability of erroneously rejecting the null hypothesis
A P value of 0.025 indicates the null hypothesis has a 5% chance of being true in a two-tailed test, True or False
False
- A statistically significant difference is determined by
The experimental design when defining α
A P-value that is equal to or smaller than α
A z-score above the critical value (in a one sided test)
T-tests enable a comparison of the means for samples which are:
Degrees of freedom and α
We cannot compare all three groups in multiple comparisons, as it will lead to…
Family wise error rate
At α = 0.05, the P(retain all three Hos)
(1−0.05)3 = 0.857, so P (reject at least one) = 1−0.847 = 0.143 - This is the family-wise error rate.
- We cannot compare all three groups in multiple comparisons, as it will lead to:
family wise error rate
what does a significance level do?
The significance level defines the distance the sample mean must be from the null hypothesis to be considered statistically significant.
what does the confidence level do?
The confidence level defines the distance for how close the confidence limits are to sample mean.
how to know if you are statistically significant?
1.If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant.
2.If the confidence interval does not contain the null hypothesis value, the results are statistically significant.
3.If the P value is less than alpha, the confidence interval will not contain the null hypothesis value.
what are confidence intervals used for?
confidence intervals to assess the precision of the sample estimate. For a specific variable, a narrower confidence interval suggests a more precise estimate of the population parameter than a wider confidence interval
Solution to multiple comparisons:
Test for overall significance using a technique called “Analysis of Variance” (ANOVA)
Do post hoc comparison on individual groups
is regression and correlation inferential or descriptive?
descriptive
is anova inferential or descriptive
inferential
is the applied to means method inferential or descriptive
inferential
what descriptive method is the bivariate and multivariate method part of, respectively?
correlation and regression for bivariate
multiple regression for multivariate
p value?
The probability that the observed test statistic is equal to or more extreme, than the observed result when Ho is true
What is the probability of the observed test statistic when Ho is true?
Probability of observed statistic is very low.
𝜇_1−𝜇_2 “ is the parameter “
True or false?
True
𝑥̄_1−𝑥̄_2 “ is the point estimator”
True
Ha: μ1 – μ2 > 0 is this left or right tailed
right
Ha: μ1 – μ2 < 0 is this left or right tailed
left
why do we not perform separate t-tests
Null hypothesis has different arrangements.
Cannot treat them as independent samples, have to compare 1, 2 and 3,not each thing individually, because end up acrewing random differences.
standard error is computed soley from sample attributes: True or False
True
