Presentation on theme: "Chapter 3 Averages and Variations"— Presentation transcript:
1 Chapter 3 Averages and Variations 3.1 Measures of Central Tendency
2 Mode, Median and MeanWhat kind of data will we be able to compute mode, median and mean?Quantitative data can have a mode, median and mean.Qualitative data can have a mode.
3 ModeThe value that occurs most frequently is the mode. Some books describe the mode as the “hump” or local high point in a histogram, which does imply frequency of an answer.
4 Median The median of a data set is the middle data value. To find, order the data from smallest tolargest, and the data set in the middle(for a data set of n, the middle positionis ) is the median.Does anyone detect a potential problem?
5 MeanYou are used to an “average” of the test. The technical term is the mean.
6 MeanYou are used to an “average” of the test. The technical term is the mean.Trimmed mean is a term for a mean where a percentage of the data values are disregarded. A 5% mean is one where 5% of top and 5% of bottom values are thrown out before computing the mean.
7 Pulse DataLets find the mode, the median and the mean of the pulse data from the first day of class.We just found the population mean (μ) rather than the sample mean ().What is the difference then between μ and ?
8 Weighted AveragesFinal Exams are computed in as weighted averages. How do they do that???
9 Weighted AveragesFinal Exams are computed in as weighted averages. How do they do that???That is, multiply the data value by its weighting, add each of those, then divide by the sum of the weighting (typically 1)
11 While knowing the mean is important There is other information from data that you can measure.These tell you about the spread of the data.Range – difference between largest and smallest value of a data distribution.
12 VarianceVariance = measure of how data tends to spread around an expected value (the mean)Each data point = xMean = Deviation = x – Sample size = nVariance = s2Standard Deviation = s
14 Variance (cont)Defining FormulaComputation Formula
15 Variance (cont)To find standard deviation, just square root the variance.The computational formula tends to be a little easier to do by hand, but we will practice both.These two formulas ARE the same.
16 Variance (cont)Lets find the variance and the standard deviation of the pulse data, using both formulas.
17 Variance (cont)If an entire population is used, instead of a sample, the notation is different but the methods are the sameEach data point = xMean = µDeviation = x – µSample size = NVariance = σ 2Standard Deviation = σ
19 Variance (cont)Coefficient of Variance (CV) expresses standard deviation as a percentage of the sample/population mean.
20 Variance (cont)Coefficient of Variance (CV) expresses standard deviation as a percentage of the sample/population mean.SamplePopulation
21 Variance (cont) Chebyshev’s Theorem For any data set, the proportion that lies within k standard deviations on either side of the mean is at leastSo 75% lies between 2 standard deviations, 88.9% between 3 standard deviations, etc.
22 3.3 Mean/Standard Deviation What if you use grouped data
23 Grouped DataLots of data = TEDIOUS, whether you have a calculator or not… If you generally approximate the mean and standard deviation, that sometimes is enoughTo deal with this, you actually begin with a frequency table (remember Histograms?
24 Grouped Data (cont) Make a frequency table Find the midpoint of each class = xCompute each class frequency = fTotal number of entries = n
25 Grouped Data (cont) Make a frequency table Find the midpoint of each class = xCompute each class frequency = fTotal number of entries = n
26 Grouped Data (cont)Defining FormulaComputation Formula
27 Grouped Data (cont)Essentially, by using the midpoint and the frequency, you use a representation for ALL data values in that class, without typing in every data value.It will be a little off, but again, if the data set is huge it isn’t a bad way to approach the problem.
29 Percentiles Baby Calculator Children’s BMI A percentile ranking allows one to know where the particular data value falls in relation to the entire population.
30 Percentiles (cont)The Pth percentile (1 ≤ P ≤ 99) is a value so that P% of the data falls at or below it (and 100 – P % falls at/above)60th Percentile does NOT mean 60% score – it means that 60% of scores fall at or below that position… 60th percentile could be 80%Where have you seen percentiles?
31 Percentiles (cont)Quartiles – special percentiles used frequently. The data is divided into fourths, called Quartiles.2nd Quartile – Median1st Quartile – Median below (exclude Q2)3rd Quartile – Median above (exclude Q2)Interquartile Range (IQR) = Q3 – Q1
32 Percentiles (cont)Lets find the quartiles for following Math class sizes in the 9th grade.10, 11, 12, 12, 14, 15, 16, 17, 19, 201st Q = 123rd Q = 17Median = 14.5IQR = 17 – 12 = 5
33 Percentiles (cont) Lets find the quartile for the pulse data Why are these values significant? These are needed to make Box and Whiskers Plots