A mathematical model is a simplification of a real world situation.

Some advantages of mathematical models are:

they are quick and easy to produce

they can simplify a more complex situation

they help us to improve our understanding of the real world as certain variables can readily be changed

they enable predictions to be made about the future

they can help provide control

Some disadvantages of mathematical models are:

they only give a partial description of the real situation

they only work for a restricted range of values

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Representation and summary of data

Variables:

Variables with numerical observations are called quantitative variables. E.g. height

Variables with non-numerical observations are called qualitative variables. E.g. hair colour

A variable that can take any value in a given range is a continuous variable. E.g. time

A variable that can take only specific values in a given range is a discrete variable. E.g.number of girls in a family

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Grouped data in a table

Adding a column of the running total of the frequencies in a table is called the cumulative frequency

with grouped data you will need to know:

the groups are commonly known as classes

you need to be able to find the class boundaries

you need to be able to find the mid-point of a class

you need to be able to find the class width

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Mode, median and mean

Mode: The mode is the value that occurs most often.

Median:

The median is the middle value when the data is put in order. If there are n observations divide n by 2.

When n/2 is a whole number find the midpoint of the corresponding term and the term above.

When n/2 is not a whole number round the number up and pick the corresponding term.

Mean:

The mean is the sum of all the observations divided by the total number of observations.

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Mode, median and mean continued

Mode:

This is used when data is qualitative, or quantitative with either a single mode of bimodal.It is not very informative if each value only occurs once.

Median:

This is used for quantitative data. It is usually used when there are extreme values.

Mean:

This is used from quantitative data and uses all the pieces of data. It therefore gives a true measure of the data. But it is affected by extreme values.

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Chapter 2 key points

A grouped frequency distribution consists of classes and their related class frequencies.

Classes 30-31 32-33 34-35

For the class 32-33

Lower class boundary is 31.5

Upper class boundary is 33.5

Class width is 33.5-31.5 = 2

Class mid-point is 1/2931.5+33.5) = 32.5

The modal class is the value or classes that occurs most often.

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Representation and summary of data

Range: range = highest value - lowest value

Finding upper and lower quartiles:

Discrete data -

lower quartile (Q1) divide n by 4.

upper quartile (Q3) divide n by four and then multiple by 3.

Continuous data -

lower quartile (Q1) divide n by 4 and then use interpolation to find corresponding value.

upper quartile (Q3) divide n by four and multiple by three and use interpolation to find corresponding value.

Interquartile range:interquartile range = upper quartile - lower quartile

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