Statistics lecture 3. Bell-Shaped Curves and Other Shapes

Statistics lecture 3 Bell-Shaped Curves and Other Shapes

Goals for lecture 3 Realize many measurements in nature follow a bell-shaped ( normal ) curve Understand and learn to compute a standardized score Learn to find the proportion of the population that falls into a given range Memorize the Empirical Rule

Histogram

Bell-Shaped Normal Curve

Remember? Mean (average): Sum of the values divided by the number of values Standard deviation: A measure of how Standard deviation: A measure of how spread out the values are. Think of it as the average distance of all values from the mean.

Some Characteristics of a Normal Distribution Symmetrical (not skewed) One peak in the middle, at the mean The wider the curve, the greater the standard deviation Area under the curve is 1 (or 100%) mean

Why it looks like that With many things in nature, most individuals fall near the average. The farther you move above or below the average, the fewer individuals there are with those extreme values. Examples: Height, weight, IQ, pulse rate

Bell-shaped wear

Not all curves are normal

Normal Curve... If you know these two things: The Mean The Standard Deviation...

...Normal Curve...you can figure these things: The proportion of individuals who fall into any range of values The percentile of any given value The value of any given percentile

Percentiles Your percentile for a particular measure (like height or IQ) is the percentage of the population that falls below you. In one of my recent classes: My height (183 cm): 89th percentile My weight (104 kg): 99 th percentile My age (62): 99 th percentile

Standardized Scores A standardized score (also called the z-score) is simply the number of standard deviations a particular value is either above or below the mean. The standardized score is: Positive if above the mean Negative if below the mean

Standardized Score Examples Class height: Mean 170 cm, StdDev: 10 cm. What is the z-score of someone: 160 cm 180 cm 175 cm 150 cm 170 cm 145 cm

Calculate z-score for a Particular Value z-score = (Value - mean) / StdDev 185 cm : (185 170) / 10 = 15 / 10 = +1.5 165 cm: (165-170) / 10 = -5 / 10 = -0.5 180 cm: (180-170) / 10 = 10 / 10 = +1.0

What s the Point? With z-score or percentile, you can compare unlike things. For instance, I am heavier (99th pctile) than I am tall (89th pctile). With a z-score, you can look up the percentile in a table or an online calculator

The Empirical Rule For any normal curve, approximately: 68% of values within one StdDev of the mean 95% of values within two StdDevs of the mean 99.7% of values within three StdDevs of the mean

Empirical Rule

Outlier A value that is more than three standard deviations above or below the mean.

Apply Empirical Rule to Class Height Class height: Mean 170 cm., StdDev 10 cm. About 68% of class is between what heights? 160 cm and 180 inches (+/- 10 cm) About 95% of class is between what heights? 150 inches and 190 inches (+/- 20 cm)

Data visualization goals See different ways of graphically displaying data. Learn the features of a good statistical picture. Be able to identify common problems with graphs and plots. Learn to read graphs comprehensively.

Why do we turn data into graphics? Easier to understand Easier to see the trends A good graphic will convey the same message you would get if you really studied the data Graphics reveal data. -- Edward Tufte

Two kinds of variables Categorical: Data that can be counted in categories, such as gender or race Measurement: Data that can be Measurement: Data that can be recorded as a number and then put into order, such as IQ, weight, cigarettes smoked per day, etc.

Pictures of Categorical Data Three common types of graphics for categorical data: Pie charts Bar graphs Pictograms

Pie Charts 37% 63% Women Men Good for showing one categorical variable, like gender Show the percentage that falls into each category

Bar Graphs Can show two or more categorical variables simultaneously (for example, height ents S tu d e 12 10 8 6 4 2 F M and gender) 0 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 Inches

Pictograms num mber A grades A B C D F Height of pictures is used like bars

Pictograms can be misleading We tend to focus on the area, rather than just the height

Pictograms can be misleading To be fair, you should keep the width of pictograms the same

Pictures of Measurement Data Lots of ways to illustrate measurement variables: Stemplots and histograms (lecture 2) Line graphs (also called fever charts) Scatter plots Others: Area, radar, doughnut, highlow-close, surface plots, maps, et al.

Stemplots 19 5 9 20 1 1 4 5 5 5 6 6 6 6 7 7 8 21 0 1 2 2 2 2 4 4 4 4 5 8 8 8 9 9 22 0 2 2 2 5 6 7 9 23 0 2 2 3 5 5 24 4 6 7 7 9 9 25 1 7

Line Graph (Fever Chart)

Scatter Plot Good for displaying the relationship between two measurement variables

Scatter Plot 350 300 Doig pounds 250 200 150 100 60 65 70 75 80 inches

Scatter Plot height vs. weight 300 pounds 250 200 150 100 60 65 70 75 80 inches

Scatter Plot height vs. w eight 300 250 pounds 200 150 100 60 65 70 75 80 inche s

Difficulties and Disasters Most common problems: No labeling on one or more axes Not starting at zero Changes in labeling on axes Misleading units Graphs based on poor information

Checklist for Statistical Pictures 1. Does the message clearly stand out? 2. Is the purpose or title evident? 3. Is a source given for the data? 4. Did the data come from a reliable, believable source? 5. Is everything labeled clearly and unambiguously?

Checklist for Statistical Pictures 6. Do the axes start at zero? 7. Do the axes maintain a constant scale? 8. Are there breaks in the numbers on the axes that may be easy to miss? 9. Have financial numbers been adjusted for inflation? 10. Is there extraneous information cluttering the picture or misleading the eye?

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