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Histograms & StemplotsHistograms & Stemplotsfor Quantitative Datafor Quantitative Data

Describing Data using Describing Data using Summary Features of Quantitative Summary Features of Quantitative

VariablesVariables

Center—Location in middle of all data

Unusual features - Outliers, gaps, clusters

Spread—Measure of variability, range

Shape—Distribution pattern: symmetric, skewed, uniform, bimodal, etc.

CUSS in context!

Dotplot for Univariate Dotplot for Univariate Quantitative DataQuantitative Data

Center: about -50

Unusual features: gap at -45

Spread: 48 degrees (-69 to -21)

Shape: trimodal

StemplotStemplot for for Quantitative Quantitative DataDataAges of Death of U.S. First LadiesAges of Death of U.S. First Ladies

3 | 4, 63 | 4, 64 | 34 | 35 | 2, 4, 5, 7, 85 | 2, 4, 5, 7, 86 | 0, 0, 1, 2, 4, 4, 4, 5, 6, 9 6 | 0, 0, 1, 2, 4, 4, 4, 5, 6, 9 7 | 0, 1, 3, 4, 6, 7, 8, 87 | 0, 1, 3, 4, 6, 7, 8, 88 | 1, 1, 2, 3, 3, 6, 7, 8, 9, 98 | 1, 1, 2, 3, 3, 6, 7, 8, 9, 99 | 79 | 7

Center: 65 years, Spread: 63 years; Center: 65 years, Spread: 63 years; Shape: skewed left (towards lower Shape: skewed left (towards lower

numbers)numbers)

Stem

Leaf—a single digit

3 | 4 indicates 34 years old

Key

How to make a Stemplot How to make a Stemplot (Stem and Leaf Plot)(Stem and Leaf Plot)

Separate each observation into a stem (all Separate each observation into a stem (all but the last digit) and a leaf (the last digit)but the last digit) and a leaf (the last digit) Sometimes rounding to the nearest hundred, Sometimes rounding to the nearest hundred,

thousand, etc. is a good idea when there are a thousand, etc. is a good idea when there are a lot of digits to considerlot of digits to consider

Write the stems in a vertical column in Write the stems in a vertical column in order from smallest to largest and draw a order from smallest to largest and draw a vertical line at the right of the columnvertical line at the right of the column

Write each leaf in the row to the right of its Write each leaf in the row to the right of its stem in increasing orderstem in increasing order

Make a stemplot with the Make a stemplot with the following datafollowing data

Joey’s first 14 quiz grades in a marking Joey’s first 14 quiz grades in a marking period were:period were:

8686 8484 9191 7575 7878 8080 747487 87 7676 9696 8282 9090 9898 9393

77 4 5 6 8 4 5 6 888 0 2 4 6 7 0 2 4 6 7 Key: 7 | 4 is score of Key: 7 | 4 is score of

747499 0 1 3 6 8 0 1 3 6 8

Center: 86; Spread: 24; Shape: UniformCenter: 86; Spread: 24; Shape: Uniform

Split StemplotSplit Stemplot

When there are very When there are very few stems, do a split few stems, do a split stemplot by splitting stemplot by splitting the stem in halfthe stem in half

For instance the 7 For instance the 7 stem is split so that stem is split so that 70-74 would go in the 70-74 would go in the first 7 stem and 75-79 first 7 stem and 75-79 would go in the would go in the second 7 stemsecond 7 stem

Joey’s scores: Joey’s scores:

77 4 4 77 5 6 85 6 88 0 2 4 8 0 2 4 88 6 76 79 9 0 1 3 0 1 3 9 9 6 8 6 8

Key: 7 | 4 is score of 74Key: 7 | 4 is score of 74

Back-to-back StemplotBack-to-back Stemplot

Babe Ruth Roger Babe Ruth Roger MarisMaris

| 0| 0 | | 8 8

| 1 || 1 | 3, 4, 6 3, 4, 6

5, 2 5, 2 | 2 || 2 | 3, 6, 8 3, 6, 8

5, 4 5, 4 | 3 || 3 | 3, 9 3, 9

9, 7, 6, 6, 6, 1, 1 9, 7, 6, 6, 6, 1, 1 | 4 | 4

9, 4, 4 9, 4, 4 | 5 || 5 |

0 0 | 6 || 6 | 1 1Number of home runs in a season

Compare Ruth & MarisCompare Ruth & Maris When comparing, use comparative language When comparing, use comparative language

(higher, better, larger, more, etc)(higher, better, larger, more, etc) Who’s Better?Who’s Better?

Babe Ruth – centered higher at about 47 compared to Babe Ruth – centered higher at about 47 compared to Maris at 23Maris at 23

Any unusual features?Any unusual features? Maris has a possible outlier at 61Maris has a possible outlier at 61

Spread?Spread? Maris has larger spread of 53 compared to Babe’s of 38Maris has larger spread of 53 compared to Babe’s of 38

Shape?Shape? Babe’s is mound shaped and symmetrical, while Babe’s is mound shaped and symmetrical, while

Maris’s is skewed right with the outlierMaris’s is skewed right with the outlier

Histogram of Discrete Data – Histogram of Discrete Data – possible values in center of barpossible values in center of bar

42

54

46 45

5954

0

10

20

30

40

50

60

70

1 2 3 4 5 6

Face of Fair Six-sided Die

Fre

qu

ency

Count

20

40

60

80

100

120

age0 20 40 60 80 100

TX_betweenHoustonDallas Histogram

Histogram of Continuous Data – possible Histogram of Continuous Data – possible values in between numbers at edges of barvalues in between numbers at edges of bar

Variable being counted

Frequency Count This bin represents the # of people whose age is at least 20 but less than 25

Centered at about 35 Skewed rightSpread of 90 years

Ch. 3 Test Results 2014Ch. 3 Test Results 2014

Centered at about 83

Gap between 50-55

Slightly Skewed left, but fairly symmetric

Spread/range of about 55

Ch. 3 Test Results 2014Ch. 3 Test Results 2014Decimal point is 1 digit(s) to the right of the colon.Leaf unit = 1 4 : 5 5 : 5 : 9 6 : 23 6 : 689 7 : 2234 7 : 56677899 8 : 0002233334 8 : 5556688889999 9 : 0001234 9 : 567710 : 00

How to make HistogramsHow to make Histograms Divide the list of data into groups or classes of Divide the list of data into groups or classes of

equal width (0-5, 5-10, 10-15, etc)equal width (0-5, 5-10, 10-15, etc) Scale the horizontal axis using these classesScale the horizontal axis using these classes Count the number of individuals in each classCount the number of individuals in each class Scale the vertical axis using the countsScale the vertical axis using the counts Draw bars so the height represents the count of Draw bars so the height represents the count of

frequency of each classfrequency of each class All bars should touch, gaps indicate a gap in the All bars should touch, gaps indicate a gap in the

datadata Ideally there should be about 5- 10 barsIdeally there should be about 5- 10 bars

Make a histogram using Joey’s Make a histogram using Joey’s scores from earlier slidescores from earlier slide

Remember a score on the border between two Remember a score on the border between two bars goes in the bar on the rightbars goes in the bar on the right

The histogram and previous stemplot should look The histogram and previous stemplot should look the same, just rotated and without individual the same, just rotated and without individual numbersnumbers

Histograms on the Histograms on the calculatorcalculator

Choose Stat, Edit and enter data into a List Choose Stat, Edit and enter data into a List Choose histogram option in StatPlotChoose histogram option in StatPlot Choose the list you used for Xlist, and 1 for Freq if all Choose the list you used for Xlist, and 1 for Freq if all

data is in one listdata is in one list Press Zoom 9 to graph Press Zoom 9 to graph Press Window and round the min, max, and xscl to Press Window and round the min, max, and xscl to

pretty values. The Xscl will set the width of the bars.pretty values. The Xscl will set the width of the bars. Press Graph to see the graph with your fixed windowPress Graph to see the graph with your fixed window Press Trace to see details of the graphPress Trace to see details of the graph