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STAT 101
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Chapter 3

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Statistics

STAT 101

Qian( Michelle) Zhou

Fall

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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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