GEA1000 chap 1 variables, mean, standard deviation, median, interquartile range

bruh im just combining the slides because there's sm shit man

what is a variable

a variable is an attribute that can be measured or labelled

define independent and dependent variables

• IV: an independent variable is a variable that may be subject to manipulation (either deliberately or spontaneously) in a study

• DV: a dependent variable is a variable which is hypothesised to change depending on how the independent variable is manipulated in a study

what are the types of variables that can be used as an IV or DV

• categorical variable

• numerical variable

what is the uses of categorical variables

• take label values

• each observation can be placed in only one label + labels are mutually exclusive

what is the uses of numerical variable

• take numerical values

• and thus arithmetic operations such as adding and averaging make sense

what are the 2 types of categorical variables and their characteristics

• ordinal

  • comes with some natural orderings

  • numbers are often used to represent the ordering (eg mood)

• nominal

  • no intrinsic ordering for their variables (eg types of animals)

what is one thing to note for ordinal categorical variables

differences between numbers are subjective and (may) not be consistent

thus, labelling categories using numbers DOES NOT transform the nature of the variable to become numerical

• calculating averages and performing arithmetic operations is not advisable

what are the 2 types of numerical variables and their characteristis

• discrete

  • possible values of the variable form a set of numbers with ‘gaps’ (eg. no of family members, no of pets in a household)

• continuous

  • can meaningfully take on all possible numerical values in a given range or interval (eg. time)

when should scatter plots be used

to model a relationship between 2 numerical values

when should histograms be used

to show a graph distribution of a single numerical value

when should bar graphs be used

to compare qualities across different categories

when should box plots be used

to compare summary statistics for a numerical variable across different categories

what should be done if the purpose of collecting the data is to get information on particular individuals

go to data set and extract the information for the particular individual(s)

what should be done if the purpose of collecting the data is to get information on groups/population

• data visualisation

• summary statistics

what is a pro and con of data visualisation

+: bring forth patterns which can be used to desc groups of individuals

-: cannot perform calculations → do summary statistics instead

what are the different summary statistics

• measures of central tendency

  • mean

  • median

  • mode

• measures of dispersion

  • standard deviation

  • interquartile range

what is the properties of the mean

• x₁ + x₂ + … + x_n = nx̄

• adding a constant value to all the data points changes the mean by that constant value

• multiplying all the values to the data points by a constant number c will result in the mean also being multiplied by c

what the mean can/cannot tell us and a misconception of the mean

eg rainfall:

can tell us:

• total rainfall for that year

  • x̄ x12 = total for the entire year

cannot tell us:

• how the rainfall is distributed across the year

misconception:

• knowing the mean ≠ 50% of the months have a rainfall of at least x̄

what is proportion

proportion is the mean for a numerical variable that only takes 2 groups

what is standard deviation

standard deviation is a way of quantifying the ‘spread’ of the data about the mean

what is the formula of standard deviation

refer to picture

why is taking the difference between each value and the mean, add up the differences to get the ‘total spread’ and then divide by the number of points to get an ‘average spread’

• if there are negative values of each positive value provided

• the difference between each value to the mean cancels each other out to become 0

what is the explicit computation of standard deviation

1. find the average value of the data set

2. subtract the average value from each of the data points and square the answer

3. add up the values in 2. and divide by n-1

4. square root the answer to get S.D

properties of standard deviation

• always non-negative

• adding constant value, c to all the data points does not change the standard deviation

  • it shifts every point by the same amount

• multiplying the data points by a constant value c results in the standard deviation being multiplied by |c| where |c| is the absolute value of c.

what is the coefficient variable

a way of quantifying the degree of spread, relative to the mean

what is the formula for coefficient variation

standard deviation / mean

what is median

the median of a numerical variable in a data-et is the middle value of the variable after arranging the values of the data-set in ascending/descending order

what is the median of an even and odd number of values

even: the average of the middle 2 values

odd: only one middle value after arranging

what are the properties of the median

• add a constant value (positive or negative) to all the data points changes the median by a constant value → shift occurs but no change in spread

• multiplying all the data points by a constant value c results in the median being multiplied by c → spread widens and shift does occur

what is the relationship between the mean and the median

when the distributions are roughly symmetric → mean and median will be quite close to one another

what is the first quartile

usually denotes as Q1 is the 25th percentile of data values

what is the third quartile

usually denoted by Q3 is the 75th percentile of the data values

what is the interquartile range

it is the difference between the third and the first quartile

IQR = Q3 - Q1

what are some similarities between the IQR and SD

• IQR and SD is always non-negative (given that Q3 is at least as large as Q1)

• adding a constant value, c to all the data points does not change the IQR and SD

• multiplying all the data points by a constant value c results in the IQR and SD being multiplied by |c|

what is the mode

the mode of a variable is the value of the variable that appears most frequently