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Representation andSummary of Data- Dispersion

Representation and Summary of Data - DispersionThe last chapter was based on calculating averages from sets of data

Just the average alone does not give the full picture though

The next chapter looks at measures of dispersion, how spread out the data isTeachings for Exercise 3ARepresentation and Summary of Data - DispersionRange and QuartilesThe Quartiles, Q1, Q2 and Q3 split the data into 4 parts, with 25% of the information in each

For discrete data Q1 = n/4 Q2 = n/2 Q3 = 3n/4For continuous data Use interpolation (like with the median from chapter 2)

Inter-quartile range Upper Quartile Lower Quartile Q3 Q1 3ALowest ValueHighest ValueQ2Q3Q125%25%25%25%Remember, if the result is whole, you need the midpoint of the term and the term above. If not, round up and find the corresponding term.

Representation and Summary of Data - DispersionRange and QuartilesCalculate the Range and Inter-quartile range of the following data.7, 9, 4, 6, 3, 2, 8, 1, 10, 15, 11

Putting the data in order1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15

Range Lower Quartile Upper Quartile Inter-quartile Range 15 1 = 14

n 411 42.75 (3rd term)33n 433 48.25 (9th term)1010 3 = 73A

Representation and Summary of Data - DispersionRange and QuartilesRebecca records the number of CDs in the collections of students in her year. The results are shown in the table opposite. Calculate the Inter-quartile range (IQR).

Q1 =

Q3 =

IQR = Q3 Q1= 38 37= 19512398334384929372017363335Cumulative FrequencyNo. Students, fxn 495 423.75 (24th term)373A3n 4285 471.25 (72nd term)38Discrete

Representation and Summary of Data - DispersionRange and QuartilesThe length of time spent on the internet each evening by a group of students is shown in the table below. Calculate the Inter-quartile range.

Q1 =

Q1 =

Q1 = 32.74 Time (mins)No. StudentsCumulative Frequency30-312232-33252734-36305737-391370n 470 417.5th termLB +(PLGFCWx)31.5 +(15.5252x)31.533.5Continuous3A

Representation and Summary of Data - DispersionRange and QuartilesThe length of time spent on the internet each evening by a group of students is shown in the table below. Calculate the Inter-quartile range.

Q3 =

Q3 =

Q3 = 36.05Time (mins)No. StudentsCumulative Frequency30-312232-33252734-36305737-3913703n 4210 452.5th termLB +(PLGFCWx)33.5 +(25.5303x)33.536.5Continuous3A

Representation and Summary of Data - DispersionRange and QuartilesThe length of time spent on the internet each evening by a group of students is shown in the table below. Calculate the Inter-quartile range.

Q1 = 32.74

Q3 = 36.05

IQR Q3 Q1 36.05 32.74 3.31

Time (mins)No. StudentsCumulative Frequency30-312232-33252734-36305737-391370Continuous3A

Teachings for Exercise 3BRepresentation and Summary of Data - DispersionPercentilesA Percentile is similar to a quartile. The 70th percentile of a set of data will be the value that has 70% of the data before it. It would normally be written P70.

The 62nd percentile will be the value that has 62% of the data before it, P62.

To calculate Px, you find the value of the th term

For the 31st percentile

For the 90th percentile

You can calculate the n% to m% Inter-percentile range Pm Pn

The Quartiles are effectively percentiles Q1 = P25 Q2 = P50 Q3 = P75

3B

Representation and Summary of Data - DispersionPercentilesThe height, in cm of 70 eighteen year old boys was measured and the data put into the table opposite. Calculate the 90th percentile, the 10th percentile and the 10% to 90% Inter-percentile range.

P90 =

P90 =

P90 = 186.67 (2dp)704190-200669180-1905732170-1802521160-17044150-160Cumulative FrequencyStudentsHeight90n 1006300 10063rd termLB +(PLGFCWx)180 +(6910x)3B

Representation and Summary of Data - DispersionPercentilesThe height, in cm of 70 eighteen year old boys was measured and the data put into the table opposite. Calculate the 90th percentile, the 10th percentile and the 10% to 90% Inter-percentile range.

P10 =

P10 =

P10 = 161.43 (2dp)704190-200669180-1905732170-1802521160-17044150-160Cumulative FrequencyStudentsHeight10n 100700 1007th termLB +(PLGFCWx)160 +(32110x)3B

Representation and Summary of Data - DispersionPercentilesThe height, in cm of 70 eighteen year old boys was measured and the data put into the table opposite. Calculate the 90th percentile, the 10th percentile and the 10% to 90% Inter-percentile range.

P90 = 186.67 (2dp)P10 = 161.43 (2dp)

The 10% to 90% Inter-percentile range P90 P10

186.67 161.43 25.24cm704190-200669180-1905732170-1802521160-17044150-160Cumulative FrequencyStudentsHeight3B

Teachings for Exercise 3CRepresentation and Summary of Data - DispersionVariance and Standard DeviationVariance and Standard Deviation are measures of how far away the data is spread from the mean. If the mean is and an observation is , then the observations dispersion from the mean is .

The variance will therefore be given by;

However, a formula which is more commonly used, especially with larger sets of data, is;

The Standard Deviation, is given by .

Sum of the squared dispersions from the mean (squaring removes any negative values)Number of observations

VarianceThe square of the meanThe mean of the squares

3C

Representation and Summary of Data - DispersionVariance and Standard Deviation

Important point: The Standard Deviation tells you the range from the mean which contains around 68% of the data (if data is normally disributed)

For example, if 100 students have a mean height of 150cm and a standard deviation of 10cm.15014016013017068 of the students are within one Standard Deviation95 of the students are within two Standard Deviations3C

Representation and Summary of Data - DispersionVariance and Standard DeviationGiven that for x;

Calculate the Variance and Standard Deviation of x.

Which part is the mean?

VarianceStandard Deviation

(2dp)3C

Representation and Summary of Data - DispersionVariance and Standard DeviationUse the formula to calculate the variance and standard deviation of the following numbers3, 4, 6, 2, 8, 8, 5218362556486484236616493x2x

Total362187

(2dp)VarianceStandard Deviation3C

Teachings for Exercise 3DRepresentation and Summary of Data - DispersionVariance and Standard Deviation from a TableAs with the averages from Chapter 2, you need to be able to calculate the Variance and Standard Deviation from a frequency table, grouped or ungrouped.3D

This was the formula from before The formula for tabled data is similar:

Variance Sum of frequency times x2 Sum of frequency times xSum of frequencyThe difference reflects the fact that each value of x will appear many times, rather than just once or a few times

Representation and Summary of Data - DispersionVariance and Standard Deviation from a TableCalculate the Variance and Standard Deviation of a set of data with the following values already calculated.3D

224873125

(2dp)VarianceStandard Deviation

Representation and Summary of Data - DispersionVariance and Standard Deviation from a TableSue records the time spent in town at lunchtime (mins) of students in her year. The results are in the table. Calculate the Standard Deviation of the time spent out of school.3D3954610142639490961292343839701107329372203261217363675105335fx2fxNo. students(f)xTotal1094096154050(3 x 35)(3 x 352)(17 x 36)(17 x 362)

4096154050109

(2dp) = 1.20

Representation and Summary of Data - DispersionVariance and Standard Deviation from a TableAndy recorded the lengths of telephone calls he made over the course of a month. Calculate an estimate of the Standard Deviation of his calls.3D756.2527.527.5125-300022.5020-25612.53517.5215-20781.2562.512.5510-15843.75112.57.5155-1025102.540-5fx2fxMid-point (x)Calls, fLengthTotal27247.53018.75(4 x 2.5)(4 x 2.52)(15 x 7.5)(15 x 7.52)

247.53018.7527

(2dp)

Teachings for Exercise 3ERepresentation and Summary of Data - DispersionCodingAs with averages, coding can be used to make data easier to work with. However, there is something extra to remember

If you have a set of data with a range of 15, and reduce every number by 2, what will happen to the range? Nothing! Range measures the spread of data, and if all the numbers are 2 less, the spread will not have changed

It is exactly the same for Standard Deviation. Because it measures the spread of data, any addition/subtraction in the coding will not need to be undone.

Any division or multiplication will have to be uncoded as normal 3E

Representation and Summary of Data - DispersionCodingUse the following code to calculate the Standard Deviation of this set of data:150, 160, 170, 180, 190

Code

15, 16, 17, 18, 19

3611932418289172561622515x2xTotal851455

(2dp)But we had divided by 10 so we must undo thisx 10

(2dp)

85145553E

Representation and Summary of Data - DispersionCodingUse the following code to calculate the Standard Deviation of this set of data:150, 160, 170, 180, 190

Code

50, 60, 70, 80, 90

810090640080490070360060250050x2xTotal35025500

(2dp)We do not need to undo as we only subtracted!

3502550053E

Representation and Summary of Data - DispersionCodingUse the following code to calculate the Standard Deviation of this set of data:150, 160, 170, 180, 190

Code

5, 6, 7, 8, 9

819648497366255x2xTotal35255

(2dp)We only need to undo the divide by 10

352555x 10

(2dp)3E

Representation and Summary of Data - DispersionCodingUse the code below to calculate the Standard Deviation of this table of data.Code

12.253.53.525120-30126217.5315-2066112.5610-150007.5125-104-4-12.540-5fy2fyyMidpoint, xCalls, fCall lengthTotal2611.534.25

(2dp)Undo the divide by 5 only

(2dp)x 5(f)(fy)(fy2)

SummaryWe have now finished chapter 3

We have seen how to calculate range and Inter-quartile range including using Interpolation from a table

We have learnt how to calculate Percentiles and Inter-Percentile range

We have calculated Variance and Standard Deviation from a table

We have also used coding to simplify calculations