Stats Maths A-Level Year 1 Edexcel

Population

The whole set of terms that are of interest.

Census

Observes or measures every member of a population

Sample

A selection of observations taken from a subset of the population which is used to find information about the population as a whole

Census advantages

It should give a completely accurate result

Census disadvantages

- Time consuming and expensive

- Cannot be used when the testing process destroys the item

- Hard to process large quantity of data

Sample advantages

- Less time consuming and expensive than a census

- Fewer people have to respond

- Less data to process that a census

Sample disadvantages

- The data may not be as accurate

- The sample may not be large enough to give information about small sub-groups of the population

Sampling units

Individual units of a population

Sampling frame

Individually named or numbered to form a list called a sampling frame

The sample should be

representative of the population

Random sampling helps to remove

bias from a sample

Random sampling methods

- Simple random sampling

- Systematic sampling

- Stratified sampling

Simple Random Sampling

A simple random sample of size n is one where every sample of size n has an equal chance of being selected

Systematic sampling

The required elements are chosen at regular intervals from an ordered list

Stratified sampling

The population is divided into mutually exclusive strata (which are proportional to the population) and a random sample is taken from each.

Stratified sampling equation

The number sampled in a stratum = (number in stratum/number in population) x overall sample size

Simple random sampling advantages

- Free of bias

- Easy and cheap to implement for small populations and small samples

- Each sampling unit has a known and equal chance of selection

Simple random sampling disadvantages

- Not suitable when the population size or the sample size is large

- A sampling frame is needed

Systematic sampling advantages

-simple and quick

-suitable for large samples and populations

Systematic sampling disadvantages

- A sampling frame is needed

- It can introduce bias if the sampling frame is not random

Stratified sampling advantages

- Sample accurately reflects the population structure

- Guarantees proportional representation of groups within a population

Stratified sampling disadvantages

- Population must be clearly classified into distinct strata

- Selection within each stratum suffers from the same disadvantages as simple random sampling

Non-random sampling methods

- Quota sampling

- Opportunity (convenience) sampling

Quota sampling

An interviewer or researcher selects a sample that reflects the characteristics of the whole population

Quota sampling advantages

- Allows a small sample to still be representative of the population

- No sampling frame required

- Quick, easy and inexpensive

- Allows for easy comparison between different groups within a population

Opportunity (convenience) sampling

The sample is taken from people who are available at the time the study is carried out and who fir the criteria you are looking for.

Quota sampling disadvantages

- Non-random sampling can introduce bias

- Population must be divided into groups, which can be costly or inaccurate

- Increasing scope of study increases number of groups, which adds time and expense

- Non-responses are not recorded as such

Opportunity (convenience) sampling advantages

- Easy to carry out

- Inexpensive

Opportunity (convenience) sampling disadvantages

- Unlikely to provide a representative sample

- Highly dependent on individual researcher

Quantitative

Quantitative data is variables that are associated with numerical observations, e.g. height

Qualitative

Qualitative data is variables that are associated with non-numerical observations, e.g. hair colour

Continuous

Any value in a given range

Discrete

Only specific values in a given range

Classes

When data is presented in a grouped frequency table, the specific data values are not shown. The groups are more commonly known as classes.

Class boundaries

The class boundaries tell you the maximum and minimum values that belong in each class

Midpoint

The midpoint is the average of the class boundaries

Class width

The class width is the difference between the class boundaries

The large data set

The weather data provided is set over two periods of time (May to October in 1987 and in 2015) and is for five UK and three overseas weather stations.

Leuchars

UK, Northern, Coastal

Leeming

UK, Northern, Inland

Heathrow

UK, Southern, Inland

Hurn

UK, Southern, Coastal

Camborne

UK, Southern, Coastal

Jacksonville

Worldwide, Northern Hemisphere, Coastal

Beijing

Worldwide, Northern Hemisphere, Inland

Perth

Worldwide, Southern Hemisphere, Coastal

Daily mean temperature

in °C - this is the average of the hourly temperature readings during a 24-hour period.

Daily total rainfall

Including solid precipitations such as snow and hail, which is melted before being included in any measurements.

tr

rainfall amounts less than 0.05 mm are recorded as 'tr' or 'trace'

Daily total sunshine

recorded to the nearest tenth of an hour

Daily mean wind direction

Mean wind directions are given as bearings and as cardinal directions.

Daily mean windspeed

in knots, averaged over 24 hours from midnight to midnight. The data for mean windspeed is also categorised according to Beaufort scale.

Beaufort scale

1: Calm, less than 1 knot

1-3: Light, 1 to 10 knots

4: Moderate, 11 to 16 knots

5: Fresh, 17 to 21 knots

Daily maximum gust

in knots - this is the highest instantaneous windspeed recorded. The direction from which the maximum gust was blowing is also recorded.

Knot

A knot (kn) is a 'nautical mile per hour'. 1 kn = 1.15 mph

Daily maximum relative humidity

Given as a percentage of air saturation with water vapour. Relative humidities above 95% give rise to misty and foggy conditions.

Daily mean cloud cover

Measured in oktas or eighths of the sky covered by cloud

Daily mean visibility

Measured in decametres (Dm). This is the greatest horizontal distance at which an object can be seen in daylight.

Daily mean pressure

Measured in hectopascals (hPa)

If you need to calculations on the large data set in your exam,

the relevant extract from the data set will be provided

Countif

Command in a spreadsheet to work out the frequency in each class

Measure of location

A single value which describes a position in a data set.

Measure of central tendency

If the single value describes the centre of the data.

Mode

The mode is the value or class that occurs most often

Median

The middle value when the data values are put in order

Mean

sum of data values over number of data values

X bar

represents the mean of the data

Frequency table mean

sum of the products of the data values and their frequencies/sum of their frequencies

Quartiles

- Lower Quartile

- Upper Quartile

Percentiles

Split the data set into 100 parts.

Lower quartile

One quarter of the way through the data set

Upper quartile

Three quarters of the way through the data set

To find the lower quartile for discrete data

divide n by 4. If this is a whole number, the lower quartile is halfway between this data point and the one above. If it is not a whole number, round up and pick this data point.

To find the upper quartile of discrete data

find 3/4 of n. If this is a whole number, the upper quartile is halfway between this point and the one above. If it is not a whole number, round up and pick this data point.

Age is always rounded

down

Interpolation

When data are presented in a grouped frequency table you can use interpolation to estimate the median, quartiles and percentiles.

When you use interpolation you are assuming the data values are

evenly distributed within each class.

Quartiles for grouped continuous data

Q1 = n/4th data value

Q2 = n/2th data value

Q3 = 3n/4th data value

For finding quartiles in an ungrouped frequency tabled

use the rules for discrete data

Range

Difference between the largest and smallest values in the data set

Measures of spread names

- Measures of dispersion

- Measures of variation

Interquartile range (IQR)

The difference between the upper quartile and the lower quartile

Q3 - Q1

Interpecentile range

The difference between the values for two given percentiles.

Variance

Each data point deviates from the mean by the amount x - ⨲

Variance =

Σx²/n - (Σx/n)²

Sxx =

Σx² - (Σx)²/n

The standard deviation is

The square root of the variance

σ =

√(Σx²/n - (Σx/n)²)

σ² in a frequency table

Σfx²/Σf - (Σfx/Σf)²

σ in a frequency table

√(Σfx²/Σf - (Σfx/Σf)²)

f in standard deviation

frequency for each group

Σf in standard deviation

total frequency

Calculate estimates for the variance and standard deviation of the data in a grouped frequency table using

the midpoint of each class interval

Sxx

summary statistic

If data is coded using the formula y = x-a/b

- The mean of the coded data is given by Ý = ⨲-a/b

- The standard deviation of the coded data is given by σy = σx/b, where ox is the standard deviation of the original data.

To find the original data from the coded data

rearrange the formulae:

- ⨲ = bÝ + a

- σx = bσy

Outlier

- Either greater than Q3 + k(Q3 - Q1)

- Or less than Q1 - k(Q3 - Q1)

Cleaning the data

The process of removing anomalies from a data set.

Anomalies

Where an outlier should be removed from the data since it is clearly an error and ti would be misleading to keep it in.

Cumulative frequency diagram

If you are given data in a grouped frequency table, you are not able to find the exact values of the median and quartiles. The diagram can help find estimates for the mean, quartiles and percentiles.