Probability Distributions

Continuous distributions

Sample size 24

Guess the mean and standard deviation

Dot plot sample size 49

Draw the population distribution you expect

Sample size 93

Sample size 476

Sample size 948

Mean 160Median 161

Standard deviation 12s

• Shape activity 1.10

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Standard Deviation 1st slide

Given a Data Set 12, 8, 7, 14, 4

The standard deviation is a measure of the mean spread of

the data from the mean.

Mean = (12 + 8 + 7 + 14 + 4) ÷ 5 = 9x

Calculate the mean

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How far is each data value from the mean?

Square to remove the negatives

Average = Sum divided by how many values

Square root to ‘undo’ the squared

(25 + 4 + 25 + 1 + 9) ÷ 5 = 12.8

Square root 12.8 = 3.58

Std Dev = 3.58

nCalculator function

Looking at distributions(simulated normal distribution)

• Small samples do not always have distributions like the population they come from.

• When looking at distributions, a sample of 30 is much too small to give a good picture of the whole population distribution.

Looking at distributions(simulated normal distribution)

• Large samples do have distributions like the population they come from.

• When looking at distributions, a sample of about 200 is sufficient to give a picture of the whole population distribution.

Estimating mean and standard deviation

To estimate mean and standard deviation, you need to know that:• The mean is pulled towards extreme values• The SD is stretched by extreme valuesIf the distribution is approximately normal, the mean is the middle, and the SD is roughly 1/6th the range (97.8% within μ ± 3σ).

Estimating mean and standard deviation for any distribution

Estimating the mean:• Estimate the median and adjust towards

extreme values.

Estimating the standard deviation:• Estimate the median distance from the mean

and adjust it (stretch it if there are extreme values).

Mean = 12.3 years SD = 1.8 years

Estimate the mean and standard deviation of the age of students completing the census@school survey.

year

12

year

9

Attr1

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Collection 1 Dot PlotWords remembered in Kim’s Game

Mean = 13.1

SD = 2.4

Mean = 9.0

SD = 2.8

Mean = 38 messages

SD = 57 messages

Text messages sent in a day by stage one university students

Mean = 10.4 pairs

SD = 8.9 pairs

Number of pairs of shoes owned by stage one university students

Mean = 5.9 words

SD = 2.5 words

Mean = 7.0 words

SD = 23 words

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words memorised with music

number of words

frequency

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word memorised without music

number of words

frequency

Continuous probability graphs

What are the units on the vertical axis for a continuous probability function?

Continuous probability graphs are

probability density functions

The vertical axis measures the rate probability/x, which is called probability density. Probability density is only meaningful in terms of area.

CONTINUOUS

Draw the probability density function for the following

bus waiting time (1)

The downtown inner link bus in Auckland arrives at a stop every ten minutes, but has no set times.If I turn up at the bus stop, how long will I expect to wait for a bus? What will the distribution of wait times look like?

a

b c

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0 10

Which is more likely: a wait of between 2 and 5 minutes, or a wait of more than 6 minutes, measured to the nearest

minute?

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0 10

Bus waiting time (2)

• My own bus route (277) runs only every half hour, and isn’t as reliable as the inner link.

• I know that the bus is most likely to appear on time, but could in fact turn up at any time between the time it is due and half an hour later.

What is the best model for wait time, given the available information?

In the real world:

Uniform models are used for modelling distributions when the only information you have are maximum and minimum.Triangular models are used for modelling distributions when the only information you have are maximum, minimum and average (could be the mode).

a

b c

What is the probability that I will have to wait longer than 20 minutes for a bus?

1 15

0 30