AIMS:

- Expose to difference measures of dispersion.
- Understand the concepts of variance and standard deviation.
- Able to calculate variance and standard deviation.

- Mean is a good summary of our data but it does not tell us how the data
grouped around the mean. Here,
**we will learn how to measure dispersion or how concentrated is the distribution of the data.** - Let us look at an example.
Height (cm) Team A Team B Team C 190 2 5 0 180 2 0 4 170 2 0 2 160 2 0 4 150 2 5 0 - One simple way to take into account of "
**concentration of data**" is to use**range.**The range is defined to be the difference between the greatest and smallest data/observations in the distribution. Refering back to the above example, Team A and Team B have the same range although we can see that their distributions are different. - Another method is to consider the
**interquartile range.**

Team A: first quartile= 160 cm, third quartile=180 and the interquartile range is 20.

Team B: first quartile= 150 cm, third quartile=190 and the interquartile range is 40.

Team C: first quartile= 160 cm, third quartile=180 and the interquartile range is 20.

These interquartile ranges provide pretty good descriptions of these data but quartiles are difficult to estimate. Moreover, estimation of quartiles using cumulative frequency curve requires us to first construct the curve and second the estimation is not very accurate. The other problem is that interquartile range basically disregard 50% of the available data. This explains why Team A and Team B have the same "dispersion measure" although we know that their distributions are different. - A standard way to take into account all the data is to use
**standard deviation**as a measure of dispersion.Aims: - You should understand this concept.
- Able to calculate the standard deviation and variance by hand.
- Able to use your calculator to obtain both standard deviation and variance.

- We will learn the relevant concepts using an example. Eight hamsters were
fed a certain diet. At the end of the month, these hamsters were weighed.
Weight changes in grammes are recorded here. Negative sign represents weight
loss.

-1, 5, 20, 15, 17, 11, 9, 36

We shall first order these data into

-1, 5, 9, 11, 15, 17, 20, 36.

The mean can be easily calculated to be 14. - One measure of despersion is called
**mean deviation.**Mean deviation as its name suggests is the mean of the deviation of an observation from the mean. Let "d= (x - )" be the deviation from the mean and "N" the total numbers of observations then

Mean deviation=__Σd__

N=__d1+ d2 +d3 + ... + dn__

NWeight Change, x -1 5 9 11 15 17 20 36 Deviation from mean , d -15 -9 -5 -3 1 3 6 22 Σd= 0 Mean deviation = __Σd__

N= 0/8 = 0 - Here is a plot of above deviations from the mean.
- The mean of deviation above is 0. In fact,
**the mean of deviation is always 0 when we have a genuine random data**because half of the data will be above the mean and the other half beneath it. If the data is not completely random then the mean of deviation is not zero. - The conclusion here is obvious. Mean of deviation is not a good measure of dispersion if we have a random data.
- The problem with the mean deviation is that when the data is genuinely
random half of the data carry negative sign (see the above diagram). To avoid
this negative sign, we could square these deviations.
This new measure of dispersion is called
**variance**.

The square root of variance is called**standard deviation.**Variance, s^{2}=Σ(x - )^{2}

N=(x1 - )^{2}+(x2 - )^{2}+(x3 - )^{2}+...+(xn- )^{2}

NStandard deviation, s=√(variance)=√Σ(x - )^{2}

N=√(x1 - )^{2}+(x2 - )^{2}+(x3 - )^{2}+...+(xn- )^{2}

N^{2}and population variance by σ^{2}. In this course, we will usually deal only with population. Thus, when using a calculator always report the population variance and standard deviation for this course.If you are using TI, for example, - then standard deviation for IB purpose is σ
- and variance = σ
^{2}.

- standard deviation for IB purpose is σ
_{n}and - variance =σ
_{n}^{2}

- Thus variance is basically the mean of the squared deviations from the mean. Standard deviation could informally be though of as the mean dispersion from the mean.
- Calculating the variance and standard deviation using the above example.
Weight Change, x -1 5 9 11 15 17 20 36 Deviation from mean , d -15 -9 -5 -3 1 3 6 22 Σd= 0 Squared of deviation, d ^{2}225 81 25 9 1 9 36 484 Σd ^{2}= 870Variance = Σ(x - ) ^{2}

N= Σd ^{2}

N= 870/8 = 108.75

Standard deviation = √(870/8)

Standard deviation ≈ 10.4 (3 s.f) - Exploration 1.

Calculate the variance and standard deviation for the following test scores in a Math class.

86, 73, 74, 66, 60, 72, 62, 75, 81.

You may want to start by constructing a table like above. First calculate the mean.

Then calculate the deviation from the mean.

Follow by square of the deviation.

variance≈63.4 (3 sf) & standard deviation≈ 7.96 (3 s.f.)Step by step practice for calculating variance and standard deviation using table.

Please enable Macro to displace the EXCEL file properly.**Download the Excel file.** - Let us know look at another example. The table below shows the number of
matches in a box of matches.
Number of matches, x 47 48 49 50 51 52 Frequency, f 3 6 11 19 12 9 - We can easily calculate the mean by using
__Σ ( fx )__

Σ f - = [3(47) + 6(48)+ 11(49)+ 19(50)+
12(51)+ 9(52) ]/ 60.

≈ 49.9667 -
x 47 48 49 50 51 52 f 3 6 11 19 12 9 Σ f = 60 deviation, d -2.96667 -1.96667 -0.96667 0.03333 1.03333 2.03333 square of deviation, d ^{2}8.80113 3.86779 0.93445 0.00111 1.06777 4.13443 f(d ^{2})26.40339 23.20674 10.27895 0.02109 12.81324 37.20987 Σ f(d ^{2}) = 109.93328

Variance ≈ 1.83 (3 s.f)

Standard deviation ≈ 1.35 (3 s.f)

In order to obtain accurate answers to 3 significant figures we have kept five decimal places throughout this calculation.

- We can easily calculate the mean by using
- Exploration 2.

These are the heights of a group of students in a grade school.Height (cm), x 146 148 150 152 156 158 Frequency, f 2 1 3 1 5 3 - Find the mean height.
- Calculate the standard deviation of the students' heights.

Please enable macro to display the page inside this table.

Step by step practice for calculating variance and standard deviation

**with frequency**using table.

Please enable Macro to displace the EXCEL file properly.**Download the Excel file.** **Summary**Given a set of data

**x1, x2, x3, x4, ..., xn**and N represents the total numbers of observations/data. ThenVariance, s^{2}=Σ(x - )^{2}

N=(x1 - )^{2}+(x2 - )^{2}+(x3 - )^{2}+...+(xn- )^{2}

NStandard deviation, s=√ (variance)=√Σ(x - )^{2}

N=√(x1 - )^{2}+(x2 - )^{2}+(x3 - )^{2}+...+(xn- )^{2}

NGiven a set of data such that

data x1x2x3...xnfrequency f1f2f3...fnThen

Variance, s^{2}=Σf(x - )^{2}

N=f1(x1 - )^{2}+f2(x2 - )^{2}+f3(x3 - )^{2}+...+fn(xn- )^{2}

NStandard deviation, s=√ (variance)=√Σf(x - )^{2}

N=√f1(x1 - )^{2}+f2(x2 - )^{2}+f3(x3 - )^{2}+...+fn(xn- )^{2}

N