STAT113 - Exam 1

Histogram

A graph that displays a frequency distribution or a relative frequency distribution

Frequency distribution (of a set of observations)

A specification of the classes (or categories, or events) into which observations are sorted, along with the number of observations in each of the classes

Frequency

Number of times the value, category, or event occurs

3 types of frequency

Cumulative, relative, cumulative relative

Cumulative frequency

Sum of the frequencies belonging to the value and the ones below it

Relative frequency

Frequency expressed as a proportion of all observations

Frequency/total # of observations

Cumulative relative frequencies

Sum of the relative frequencies and the ones below it

How to make a histogram

1. Break up the range of data values into classes and count number of observations that fall into each class

   1. Initial class boundary = left endpoint of the first class

   2. Class width = width of the interval defining the class

2. Classes are displayed on the horizontal axis, while class frequencies are displayed on the vertical axis.

Peak

Bar or small group of bars that, relative to the bars nearby, is a clear and prominent high point of the histogram

Unimodal

Has one peak

Bimodal

Has two peaks

Symmetric

Left half is a mirror image of the right half

Uniform

All bars are the same height

Bell shaped

Looks like bell, unimodal and symmetric

Tail

Has a tail if one side of the histogram bars tend to get shorter and shorter the farther they are from the peak

Skewed to the left/negatively skewed

Has a long tail that extends left

← Peak

Skewed to the right/positively skewed

Has a long tail that extends to the right

Peak →

Quantitative variable

Values are numbers representing amounts

Ex. temperature, birth year

Categorical (or qualitative) variables

One whose values are categories, or “kind” or “type”

Ex. US state of current residence, eye color

Nominal categorical

No natural ranking or ordering

Ordinal categorical

DOES have natural ranking or ordering

Interval level

Quantitative data

Differences between values are meaningful, no meaningful 0 value

Ratio level

Quantitative data

Differences between values are meaningful, 0 has a meaningful value

Population

The collection of all objects under study

Parameter

A measurement obtained from every object of a population, characteristic of a population

Sample

Smaller collection of objects from the population

Statistic

A measurement obtained from a sample, usually given as a number

What is the key difference between a parameter and a statistic?

A statistic is always knowable, while a parameter may be impractical or impossible to know

Discrete variables

Quantitative data

One whose possible values can be counted, meaning that its possible values correspond to some subset of the whole numbers

Often arise when the quantities are being counted (“The number of…”)

Continuous variable

Quantitative data

One whose values make up an interval of real numbers, could take any value in a range of numbers

Often arise when the quantities are being measured

Random sample

A sample of predetermined size is chosen from the population, care is taken in such a way that every sample of that size is equally likely to be selected

Systematic sample

The members of a population are listed in some order, then every (x)th member is selected for the sample until a predetermined sample is reached

Stratified sample

Population divided into groups, with each group made up of members that share some particular characteristics, e.g. an occupation, then a fixed number of members (not all members) from each group are randomly selected for the sample

Cluster sample

Population is divided into groups that tend to exist naturally, then all members of some randomly chosen groups (but not all groups) are selected for the sample

Convenience sample

Members of the population that are easily accessible are selected for the sample

Experimental units

The people, objects, or other entities being measured

Response variable/dependent variable

The aspect being measured for each experimental unit

Explanatory variable/independent variable

The variable that is manipulated by the researcher

Observational studies

The researcher simply observes what is currently happening, or what has happened in the past, and uses the observations to try and make conclusions, does not actively impose treatments on the experimental units

Prospective study

Type of observational study

Participants are identified in advance, and data are collected over time as events unfold

An observational study cannot be both prospective and retrospective.

Retrospective study

Type of observational study

The data are collected after the outcome has occurred, using records obtained from the past

An observational study cannot be both prospective and retrospective.

Designed experiments

The researcher manipulates conditions to try to determine the effect of the conditions on the response variable, researcher randomly assigns experimental units to treatments

Double-blind experiment

Type of designed experiment

Neither the person interacting with the participants nor the participants know which participants have been assigned to which treatment

A designed experiment can be both double-blind and completely randomized.

Completely randomized experiment

Type of designed experiment

Participants are randomly assigned to groups, and then the groups are randomly assigned to receive treatments

A designed experiment can be both double-blind and completely randomized.

What are the two reasons that an observational study would be done instead of a randomized experiment?

1. It may be unethical to assign participants to treatments they are opposed to, or that may harm them

2. Researcher may not have control over the treatments

Confounder

A variable whose effect on the outcome cannot be distinguished from the effect of the on the outcome from different treatments

What are the two methods to eliminate confounders?

1. Blocking, = participants who are known to be similar in some way are grouped together

2. Matched pairs

Sampling bias

When the researcher chooses a sample in such a way that certain members are more likely to be included than others

This can be avoided by taking a random sample.

Voluntary response bias

When the members of the sample of the population voluntarily choose to participate.

Bias comes from the tendency for volunteers to not be representative of the whole population.

Researcher does not choose the sample.

Response bias

The way that the study is conducted causes the members of the sample to give inaccurate or untruthful responses.

Can be reduced by allowing respondents to answer anonymously and wording questions with a clear and neutral response

Nonresponse bias

When a study has many people that do not respond

Can be reduced by offering an incentive to respond or sending reminders to those who have not yet responded

Is standard deviation of a population different than standard deviation of a sample?

Yes.

Standard deviation of a population

The measure of the “spread” of the measurements of the population about their mean.

The more that measurements tend to differ (deviate) from their mean, the greater the standard deviation.

How to find standard deviation of a population

1. Find their mean.

2. Compute the average of their squared deviations from the mean: (x-mean)² + (x-mean)² etc. / # of observations

3. This gives the population variance, which is standard deviation squared, so you must take the square root of the number calculated

How to find standard deviation of a sample

1. Find the mean.

2. Compute the average of their squared deviations from the mean: (x-mean)² + (x-mean)² / # of observations - 1

3. Take the square root of number found.

Mean

The sum of the measurements in a data set/number of measurements in the data set

Median

The “middle” value when the data have been ordered

Mode

The measurement (or measurements) that appear with the highest frequency

mu (μ)

Represents the mean of a population

sigma (σ)

Standard deviation of a population

sigma (σ) squared

Variance of a population

x with a bar over it

Mean of a sample

s

Standard deviation of a sample

s squared

Variance of a sample

How to approximate the mean of a data set given a frequency distribution

1. Find the midpoint of each class - add the class limits (smallest and largest numbers of each class) and divide them by 2

2. Multiply each classes frequency by the midpoint

3. Add up all those products

4. Divide by the size of the data set

How to approximate the mean of a data set given a histogram

Estimated mean = + (frequency/sum of frequencies) * (midpoint) +/total number of frequencies

Range

The difference between the largest and smallest values

Five-number summary

1. Minimum

2. Lower quartile

3. Median

4. Upper quartile

5. Maximum

Interquartile range (IQR)

Upper quartile - lower quartile

How to find lower quartile of a data set

1. Multiply number of data values * 0.25

2. Round up to nearest integer

3. That is the number in the ordered data set that is the lower quartile

How to find upper quartile of data set

1. Multiply number of data values * 0.75

2. Round up to nearest integer

3. That is the number in the ordered data set that is the lower quartile

When is the median the better measure for a data set?

When the data set has outliers.

When is the mean the better measure for a data set?

When the data set doesn’t have outliers.

When is the mode the better measure for data set?

When categorical data is being used.

What does small range indicate about a data set?

Low variability in data.

What does a large range indicate about a data set?

High variability in data.

What does small mean indicate about a data set?

The values in the data set are low on average.

What does large mean indicate about a data set?

The values in a data set are high on average.

Box-and-whisker plots

Using IQR to find outliers in a data set

1. Calculate upper and lower quartile.

2. Find IQR.

3. For lower boundary, subtract 1.5*IQR from the lower quartile. For the upper boundary, add 1.5*IQR to the upper quartile.

4. Whatever number in the data set are above or below these numbers are outliers.

Approximating the standard deviation of a data set given a histogram

1. Find weighted mean, = (+ (frequency/total number of observations)*(midpoint) +)/size of data set

2. (+ frequency*(midpoint-estimated mean)²)/(size of data set - 1)