MEDD8815 Chap.1 Distributions, Summary Statistics and the Normal Curve
Distributions, Summary Statistics and the Normal Curve
What the first session asks you to be able to do
The opening session of the course treats statistics as the science of learning from data and starts where every analysis starts: with a single variable. You learn to say what kind of variable you are looking at, to draw the graph that suits it, to describe its shape, centre and spread, and to decide which numerical summaries are honest for that shape.
The session closes with the Normal distribution, which returns in every later session as the reference curve for inference.
Measurement levels decide what you may compute
The lecture slides extend the textbook's split between categorical and quantitative variables into four measurement levels.
Nominal variables name categories, ordinal variables put categories in order, interval variables have meaningful differences, and ratio variables also have a true zero. The practical consequence is immediate in software: a category stored as a number can be averaged by SPSS, and the result is meaningless.
The chapter uses variables from the survey file that runs through the course handouts to make the point concrete.
From graph to summary
Pie charts and bar graphs display categorical distributions; histograms and stemplots display quantitative ones. Every description names the overall pattern and the deviations from it.
The chapter then builds the two standard pairs of summaries, the mean with the standard deviation and the median with the interquartile range, and explains when each is appropriate.
A worked data set with one long right tail shows the mean pulled towards the tail while the median stays put, and a boxplot shows the 1.5 times IQR fence flagging exactly one suspected outlier.
Density curves and Normal calculations
A density curve is an idealised histogram with total area one.
The Normal curves form the most important family, each fixed by its mean and standard deviation, and the 68-95-99.7 rule gives a quick sense of spread. Any Normal question becomes an area question once values are standardised, and Table A answers it in either direction: from a value to a proportion, or from a proportion back to a value.
The chapter ends with a drill page on the four skills the first homework returns to most often, and with the SPSS Explore route that prints the descriptive statistics, a stemplot, a boxplot and a Normal quantile plot together.
What this chapter covers
- 01
Cases, variables and the four measurement levels
- 02
Bar graphs, pie charts, histograms and stemplots
- 03
Shape, centre, spread and outliers
- 04
Mean and median, and why they disagree in skewed data
- 05
Quartiles, the five-number summary and boxplots
- 06
The 1.5 times IQR rule for suspected outliers
- 07
Standard deviation and linear transformations
- 08
Density curves, Normal distributions and Table A
Worked example · free
Centre, spread and one suspicious value
- 2Order the data and read off the median and the two quartiles.
- 2Compute the IQR and both fences, and test the largest value.
- 2Choose the summaries that suit the shape and justify the choice.
Key terms
- Categorical Variable
- A variable that sorts each case into one of a set of groups or categories. Nominal categories have no order; ordinal categories have a meaningful order.
- Quantitative Variable
- A variable taking numerical values for which adding and averaging make sense. Interval variables have meaningful differences, and ratio variables also have a true zero.
- Median
- The midpoint of the ordered data, with half the observations below it and half above. It is resistant to extreme values.
- Interquartile Range
- The distance from the first quartile to the third quartile, covering the middle half of the data; written IQR = Q3 minus Q1.
- Standard Deviation
- A measure of spread about the mean, found as the square root of the sum of squared deviations divided by n minus 1. It is not resistant.
- Density Curve
- A curve on or above the horizontal axis with total area exactly one beneath it, so that the area over any interval is a proportion of observations.
- Standardised Value
- The number of standard deviations an observation lies from the mean, z = (x minus the mean) divided by the standard deviation.
Distributions, Summary Statistics and the Normal Curve FAQ
When should I report the median instead of the mean?
Use the median together with the interquartile range whenever the distribution is clearly skewed or has outliers, because both are resistant. Use the mean with the standard deviation only for roughly symmetric data without outliers. Plotting the data first is how you tell which case you are in.
How do I find a value from a percentile on a Normal curve?
Look inside the body of Table A for the area closest to the percentile, read the z value at the margins, and unstandardise with x = mean plus z times the standard deviation. For the tenth percentile the table gives z close to minus 1.28.
Why does the standard deviation divide by n minus 1?
The deviations from the sample mean always add to zero, so only n minus 1 of them are free to vary once the mean is known. Dividing by n minus 1 corrects for this, and it is the version SPSS and the course use throughout.
Assessment move
Before the second class, take any small data set and do everything by hand once: order it, find the five-number summary and fences, compute the standard deviation in a table of deviations, then check each number with SPSS Explore. The first quiz tests this material directly, and doing it by hand once makes every later SPSS output readable.
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