Chapter 3: Frequency Distributions
In Chapter 3: 3.1 Stemplots 3.2 Frequency Tables 3.3 Additional Frequency Charts
Start by exploringthe data with Exploratory Data Analysis (EDA) A popular univariate EDA technique is the stem-and-leaf plot The stem of the stemplot is an number-line (axis) Each leaf represents a data point Stemplots You can observe a lot by looking – Yogi Berra
Stemplot: Illustration • 10 ages (data sequenced as an ordered array) 05 11 21 24 27 28 30 42 50 52 • Draw the stem to cover the range 5 to 52: 0| 1| 2| 3| 4| 5| ×10 axis multiplier • Divide each data point into a stem-value (in this example, the tens place) and leaf-value (the ones-place, in this example) • Place leaves next to their stem value • Example of a leaf: 21 (plotted) 1
Stemplot illustration continued … • Plot all data points in rank order: 0|5 1|1 2|1478 3|0 4|2 5|02 ×10 • Here is the plot horizontally 8 7 4 25 1 1 0 2 0------------0 1 2 3 4 5------------Rotated stemplot
Shape Central location Spread Interpreting Distributions
Shape • “Shape” refers to the distributional pattern • Here’s the silhouette of our data X X X X X X X X X X ----------- 0 1 2 3 4 5 ----------- • Mound-shaped, symmetrical, no outliers • Do not “over-interpret” plots when n is small
Shape (cont.) Consider this large data set of IQ scores An density curve is superimposed on the graph
Examples of Symmetrical Shapes
Examples of Asymmetrical shapes
Modality (no. of peaks)
Kurtosis (steepness) fat tails Mesokurtic (medium) Platykurtic (flat) skinny tails Leptokurtic (steep) Kurtosis is not be easily judged by eye
Gravitational center ≡ arithmetic mean “Eye-ball method” visualize where plot would balance on see-saw “ around 30 (takes practice) Arithmetic method = sum values and divide by n sum = 290 n = 10 mean = 290 / 10 = 29 Gravitational Center (Mean) 8 7 4 25 1 1 0 2 0------------0 1 2 3 4 5 ------------ ^ Grav.Center
Central location: Median • Ordered array: 05 11 21 24 27 28 30 42 50 52 • The median has depth (n + 1) ÷ 2 • n = 10, median’s depth = (10+1) ÷ 2 = 5.5 • → falls between 27 and 28 • When n is even, average adjacent values Median = 27.5
For now, report the range (minimum and maximum values) Current data range is “5 to 52” The range is the easiest but not the best way to describe spread (better methods described later) Spread: Range
Stemplot – Second Example • Data: 1.47, 2.06, 2.36, 3.43, 3.74, 3.78, 3.94, 4.42 • Stem = ones-place • Leaves = tenths-place • Truncate extra digit (e.g., 1.47 1.4) |1|4|2|03|3|4779|4|4(×1) • Center: median between 3.4 & 3.7 (underlined) • Spread: 1.4 to 4.4 • Shape: mound, no outliers
Third Illustrative Example (n = 25) • Data: 14, 17, 18, 19, 22, 22, 23, 24, 24, 26, 26, 27, 28, 29, 30, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38 • Regular stemplot: |1|4789|2|223466789|3|000123445678×10 • Too squished to see shape
Third Illustration; Split Stem • Split stem-values into two ranges, e.g., first “1” holds leaves between 0 to 4, and second “1” will holds leaves between 5 to 9 • Split-stem |1|4|1|789|2|2234|2|66789|3|00012344|3|5678×10 • Negative skew now evident)
How many stem-values? • Start with between 4 and 12 stem-values • Then, use trial and error using different stem multipliers and splits → use plot that shows shape most clearly
Fourth Example: n = 53 body weights Data range from 100 to 260 lbs:
Data range from 100 to 260 lbs: • ×100 axis multiplier only two stem-values (1×100 and 2×100) too few • ×100 axis-multiplier w/ split stem 4 stem values might be OK(?) • ×10 axis-multiplier 16 stem values next slide
Fourth Stemplot Example (n = 53) 10|0166 11|009 12|0034578 13|00359 14|08 15|00257 16|555 17|000255 18|000055567 19|245 20|3 21|025 22|0 23| 24| 25| 26|0 (×10) Shape: Positive skewhigh outlier (260) Central Location: L(M) = (53 + 1) / 2 = 27 Median = 165 (underlined) Spread: from 100 to 260
Quintuple-Split Stem Values 1*|0000111 1t|222222233333 1f|4455555 1s|666777777 1.|888888888999 2*|0111 2t|2 2f| 2s|6 (×100) Codes for stem values: * for leaves 0 and 1 t for leaves two and threef for leaves four and fives for leaves six and seven. for leaves eight and nine For example, 120 is:1t|2(x100)
SPSS Stemplot, n = 654 Frequency counts Frequency Stem & Leaf 2.00 3 . 0 9.00 4 . 0000 28.00 5 . 00000000000000 37.00 6 . 000000000000000000 54.00 7 . 000000000000000000000000000 85.00 8 . 000000000000000000000000000000000000000000 94.00 9 . 00000000000000000000000000000000000000000000000 81.00 10 . 0000000000000000000000000000000000000000 90.00 11 . 000000000000000000000000000000000000000000000 57.00 12 . 0000000000000000000000000000 43.00 13 . 000000000000000000000 25.00 14 . 000000000000 19.00 15 . 000000000 13.00 16 . 000000 8.00 17 . 0000 9.00 Extremes (>=18) Stem width: 1 Each leaf: 2 case(s) 3 . 0 means 3.0 years Because nlarge, each leaf represents 2 observations
Frequency Table AGE | Freq Rel.Freq Cum.Freq. ------+----------------------- 3 | 2 0.3% 0.3% 4 | 9 1.4% 1.7% 5 | 28 4.3% 6.0% 6 | 37 5.7% 11.6% 7 | 54 8.3% 19.9% 8 | 85 13.0% 32.9% 9 | 94 14.4% 47.2%10 | 81 12.4% 59.6%11 | 90 13.8% 73.4%12 | 57 8.7% 82.1%13 | 43 6.6% 88.7%14 | 25 3.8% 92.5%15 | 19 2.9% 95.4%16 | 13 2.0% 97.4%17 | 8 1.2% 98.6%18 | 6 0.9% 99.5%19 | 3 0.5% 100.0%------+-----------------------Total | 654 100.0% • Frequency≡ count • Relative frequency≡ proportion • Cumulative [relative] frequency≡proportion less than or equal to current value
Class Intervals • When data sparse, group data into class intervals • Classes intervals can be uniform or non-uniform • Use end-point convention, so data points fall into unique intervals: include lower boundary, exclude upper boundary • (next slide)
Class Intervals Freq Table Data: 05 11 21 24 27 28 30 42 50 52
Histogram For a quantitative measurement only. Bars touch.
Bar Chart For categorical and ordinal measurements and continuous data in non-uniform class intervals bars do not touch.
