2.1 Density Curves & the Normal Distribution

REVIEW: To describe distributions we have both graphical and numerical tools.  Graphically: histograms, stemplots, bar graphs, circle graphs, timeplots, dotplots, boxplots.  Numerically: The five-number summary, Center (IQR), and spread (mean and standard deviation).

Density Curves If we have a distribution with one peak in the middle with the other values tapering off on the two sides almost uniformly, in other words the display looks almost symmetrical, we can describe this data with a smooth curve called a density curve. (Pg. 78)

Density Curve  Is an idealized description of the data ignoring an outliers or minor irregularities.  Is always graphed above the x – axis  The area underneath the curve is exactly 1 or 100%.

Find the proportion under the curve for the following range: A. 12 ≤ X ≤ 22B. 16 ≤ X ≤ 22 C. 16 ≤ X ≤ 18D. 13 ≤ X ≤ 21 A selected area under the curve gives the proportion of observations that fall within that range X

 and  To distinguish the mean and standard deviation of density curves from those of the actual observations, we use mu, , (Greek lowercase letter) for the mean and sigma, , (another lowercase Greek letter), for the deviation. These are known as population parameters.

Key Terms  Median – middle value in our collection. An equal number of values are above and below the median also referred to as the equal-areas point.  Mean – balance point in our curves.  Quartiles – split the area in fourths.  Mode – peak of our density curves.

When a density curve is symmetric, the Mean and Median are the same.

When the curve is skewed right, the mean lies to the right of the median. Why?

When the curve is skewed left, the mean lies to the ________ of the median. Why? Left Mean is nonresistant and is pulled in the direction of the extremely small number(s) even though we could possibly have more larger numbers.

Assignment Exercises 2.1 – 2.4 Homework Help slides (HHS)HHS

Normal Distributions When the mean and the median are the same, we say that the curve is bell-shaded. The mean is at the center of the data, from the peak down to the horizontal axis. The standard deviation is located at the inflection points (where there is a change in the curve). Notation: N(, )

Empirical Rule Rule The mean and standard deviation are the main identifiers for normal distributions.  68% of the observations will lie within one standard deviation of the mean  95% of the observations will lie within two standard deviations of the mean  99.7% of the observations will lie within three standard deviations of the mean.

Example The height of a giraffe can be described by a normal distribution. Mean,  = 204 inches with a standard deviation,  = 5.5 inches. Draw a normal curve to represent this information.

Example continued  What percent of giraffes are taller than 215 inches?  What percent of giraffes are shorter than 204 inches?  What percent of giraffes are taller than inches?  A height of inch corresponds to what percentile of giraffe height?

Think and Write A student states that he got a 640 on both the verbal and math sections of the SAT. The student stated, “I feel stronger in math, so I don’t understand what went wrong.” After researching the College Board’s website the following was found: Math:  = 455,  = 100 Verbal:  = 300,  = 85. Does this imply that the math score is actually better or worse? Explain.

Exercises & Extra Credit (EC)  Exercises: 2.5 – 2.9. Read through the summary on Page 90. Complete exercises 2.12 and  Extra Credit: Complete Exercise 2.17: FLIP50 and submit your findings by __________.