# Unit5_NormalDist_Part1 - STAT 1000 A02 Winter Term 2010...

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1STAT 1000 A02 Winter Term 2010………………………………………………………………………………………………………………………………………………………Density Curves and Normal Distributions………………………………………………………………………………………………………………………………………………………This note refers to Section 1.3 of Chapter 1 inthe text.Before we start normal distribution, we first talkabout something called density curves. We nowknow how to draw a histogram. Now we candraw a smooth curve though the tops of thehistogram bars, which will give us a gooddescription of the overall pattern of the datadistribution.
2For example, consider Figure 1.24a (page 54)and Figure1.24b (page 54) in the text:The above smooth curves give us a gooddescription of the overall pattern of the data ofthe Rainwater pH values and Survival timesrespectively. The curves are the mathematicalmodels for the distributions.
3Density Curve:A Density Curve is a curve that is always on orabove the X-axis and has area exactly 1underneath it.A Density Curve describes the overall pattern ofa distribution. The area under the curve andwithin any range of values is the proportion ofall observations that fall in that range.
4Mean and Median of a Density CurveFor a density curve, themedianis the point onthe X-axis with area 0.5 under the curve to eachside. That is, the median is the point that dividesthe area under the curve in half.Themeanof density curve is mathematicallymore complicated but can be considered as thebalance point, at which the curve would balanceif made of solid material.
5For a symmetric density curve, the mean and themedian are equal, and they both lie at the centreof the curve.
6The mean of a skewed curve is pulled away fromthe median in the direction of the long tail.
7Questions!Identify the points A, B, C (in terms of mean,median and mode) in Figure 1.38 (page-73) ofthe text!
8Normal DistributionsAs we noted earlier, when we draw a graph ofour data, we try to reveal some patterns orshapes, which will give us insight about the data.One very common pattern that might be seen ina histogram, for example, is the so calledBellShape.The appearance of such a shape may indicatethat our data is following what is called theNORMAL DISTRIBUTION.
9Normal distribution of a variable X is describedby giving its meanμand its standard deviationσ.Notation:X ~ N (μ,σ)We say that the variable X follows a normaldistribution with meanμand standard deviationσ.Changingμwithout changingσmoves thenormal curve along the horizontal axis (X-axis)without changing its spread.Changingσwithout changingμmoves thenormal curve along the vertical axis (Y-axis)without changing its mean.
10The 68-95-99.7 RuleIn a normal distribution with meanμandstandard deviationσ, it follows that68% of the observations fall withinσof themeanμ.95% of the observations fall within 2σof themeanμ.99.7% of the observations fall within 3σofthe meanμ.μ-3σμ-2σμ-σμμ+σμ+2σμ+3σ
11Standardizing and z-scoresAll normal distributions share many commonproperties. They are the same if we measure inunits of sizeσ

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