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Chapter 3

Chapter 3.odt

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STAT 101
Qian( Michelle) Zhou

Chapter 3 – The Normal distributions Exploring a distribution • 1. always plot your data: make a graph, usually a histogram or stemplot • 2. look for overall pattern (shape, center, spread) and for striking deviations such as outliers • 3. Calculate a numerical summary to briefly centre and spread • 4. Sometimes overall pattern of a large number of observations is so regular that we can describe it by a smooth curve Density Curves • Density curve is.. ◦ always on or above the horizontal axis, never negatives ◦ has area exactly 1 underneath it ◦ are idealized patterns, symmetric ◦ describes overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range ◦ shows a single high peak as a main feature of distribution ◦ good description of the overall pattern of distribution ▪ however outliers are not described by density curve Describing Density Curves • area under a density curve represents proportions of the total number of observations ◦ median of a density curve is the equal-areas point ◦ the mean is the point at which the curve would balance if made of solid material ▪ aka balance point • see-saw balance ▪ symmetrical curve --> median and mean are the same ▪ if skewed curve, mean is pulled away from median in direction of the long tail • in a density curve, we need to distinguish mean and SD, so ◦ mean = μ (mu) ◦ standard deviation = σ (sigma) Normal Distribution • normal curves – have same overall shape: symmetric, single-peaked, bell-shaped ◦ distributions they describe are called normal distributions ▪ Normal distribution – completely specified by two numbers, it means mean (mu) and SD (sigma) ▪ SD is distance from centre to change of curvature points on either side • mean located at centre of symmetrical curve, and is same as median ◦ changing μ w/o changing σ moves the Normal curve along the horizontal axis w/o changing its spread ◦ SD controls σ controls spread of a normal curve. Curves with larger SD = more spread out • point at which change of curvature takes place at distance σ on either side of μ ◦ mu and sigma alone don't specify shape of most distribution • why important? ◦ 1. good descriptions for some distribution of real data – distribution often close to Normal (such as exam scores, repeated careful measurements of same quantity, biological population etc) ◦ 2. good approx to results of many kinds of chance outcomes (ex. Proportion of heads in many tosses of a coin) ◦ 3. many statistical inference procedures based on ND work well for other roughly symmetrical distribution • many income distributions are skewed to right The 68-95-99.7 Rule • Normal Distribution w/ mean mu and sd sigma: ◦ approx 68% of observations fall within σ or mean μ ◦ approx 95% of observations fall within 2σ or μ ◦ approx 99.7% of obs. Fall withing 3σ or μ ex) graph on pg 78; find the scores that 95% of all scores are between, and suppose that the mean is 6.84 and sd is 1.55 μ - 2σ = 6.84 – 2(1.55) = 3.74 μ + 2σ = 6.84 + 2(1.55) = 9.94 • we use normal distribution b/c it is a good approximation, better than stopping at tenths ◦ better at describing real data better in teh centre of distribution rather than extreme high and low tails ex) 3.3 Iowa test scores on pg 79 ~Score 5.69 is one standard deviation from mean (6.84). What % of scores are higher than 5.29? 5.29 – 8.39 is 68%, meaning the the sum of the rest of the unshaded parts is 32. Since it's symmetrical, you can take 32 and divide it by 2 to find half of the unshaded parts, which is 16%. Now take 16+68 = 84% of the scores are higher than 5.29 • short notation for Normal distribution is N(μ , σ) The Standard Normal Distributions • all normal distributions are same if mean (mu) is centre and SD (sigma) as unit size ◦ changing these units is called standardizing ◦ to “standardize”, we subtract mean of the distribution, then divide by standard deviation ◦ standardize value = z-score ▪ z = ( x - μ )/ σ ▪ z-scores tells us how many SD the original observation falls away from mean and in which direction • Observations > mean = positive, observations < mean = negative ex 3.4) Standardizing women's heights ~ Women ages 20-29 are approx Normal w/ mean = 63.3 inches and SD = 2.7 inches Standardized height is z = height – 64.3/2.7 ~Awoman that is 70 inches tall for example has a standardized height of z = 70-64.3/2.7 = 2.11 standard deviation above the mean height ~Another example of woman who is 60 inches z =
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