MDM4U: Unit 3 - One Variable Statistics Test

Frequency

the number of times a value for a variable occurs in a data set.

Frequency Table

a table listing a variable together with the corresponding frequency of each value. It can be used for numerical or categorical data.

Stem & Leaf Plot

A way to organize a collection of numbers. Particularly useful for a data set that is large and has a large range of values. Useful for numerical data only.

Relative Frequency

the frequency of each class as a proportion/percent of the entire data set. The number of items divided by the total.

culmulative frequency

the total/sum of all the frequencies up to and including a certain value of the variable in an ordered set of measures.

Frequency table

Best for organizing

Stem and leaf plot

Best for trends

Mean

the sum of all variables in a data set, divided by the number of values

Median

the middle value of an ordered data set

Mode

value that occurs the most frequently in a data set

Outlier

a value in a data set that is very different than all others.

Outliers

will affect the mean the most

Outliers

Use the median to avoid misrepresenting the data.

mean < median < mode

Left skewed

mean = median = mode

symmetrical

mean > median > mode

right skewed

Quartiles

divide an ordered data set into four equal sized groups

divided into - lower quartile (Q1), the median (Q2) & upper quartile (Q3)

Interquartile Range (IQR)

Contains the middle 50% of the data


Calculated using Q3 - Q1

The larger the interquartile range

the larger the spread of the middle portion of the data

Semi-Interquartile Range

to help us determine if the data set is symmetrical or skewed

Box-and-whisker plot

illustrates these measures of: Q1, Median, Q3, IQR and Range

Data contains outlier

use modified box-and-whisker plot

Outlier

if value is

> Q3 + 1.5IQR

Outlier

if value is

< Q1 - 1.5IQR

Data is Symmetric

Median is centred to the box plot

Whiskers are equal length

S-IQR will be equal

Left Skewed

- box is located further to the right

- left whisker is generally longer

- median is further to the right of the box

- Q2 - Q1 > Q3 - Q2

Right skewed

- box located further left

- right whiskers generally longer

- media further to the left in the box

- Q2 - Q1 < Q3 - Q2

Five number summary

useful in descriptive analyses or during the preliminary investigation of a large data set.

Five number summary

- Median identifies the centre of a data set.

Five number summary

upper and lower quartiles span the middle half of a data set

Five number summary

highest and lowest observations provide information about the dispersion of data

Percentiles

divide an ordered data set into 100 equal sized groups

z-SCORES

the number of standard deviations that a data point is from the mean

Range

the difference between the largest and smallest score in the data set

it indicates how spread out the data is from two extreme values in the data set

Deviation

the difference (distance) between an individual value in a data set and the mean for the data

Deviation

= value - mean

Average Deviation

Sum all dev / number of values

Larger the size of the deviation

the greater the spread in the data

Variance

the average of the squared deviatioins.

Variance

Take all the deviations of an entire data set, square each deviation and find the means of these values

Variance Formula

Standard Deviation

measure of spread that approximates the standard/typical distance each value is from the mean

Standard Deviation Formula

the square root of the variance

- go to formula

Standard Deviation for entire population formula

Standard Deviation for sample population formula

Calculating Variance

1. Calculate the mean of the data set

2. Calculate the deviation for each data point, and then square them

3. Calculate the average of these squared deviations

Calculating Standard Deviation

1. Calculate the Variance

2. Calculate the square root of the variance

Coefficient of variation

refers to a statistical measure of the distribution of data points in a data series around the mean.

Coefficient of variation

Represents the ratio of the standard deviation to the mean.

Coefficient of variation

helpful in comparing the degree of variation from one data series to the other, although the means are considerably different from each other

Coefficient of variation

is a measure of spread that describes the amount of variability relative to the mean.