Statistics & DataThe standards cover three topics:

1. Use dot plots, histograms, stem plots, and box plots.

2. Use appropriate stats to compare center and spread.

3. Interpret differences from data sets in the shape, center, and spread accounting for outliers.

Stem Plots (or Stem and Leaf)

Be able to READ them!!!

The "stem" gives you the tens/hundreds place and the "leaf" gives you

the ones place.

Write out the numbers for the data from the stem plot.

1,1,2,2,3,4,4,4,4,5,8,10,10,10,11,11,13,17,19,25,25,27,27,28,28,29,29,30,31,31,31,32,32,32,34,35,40,44,48,49,52,56,57,57,58,63,66

Double Stem­and­Leaf Plot

Write out the data for the women's team.

65, 68, 69, 72, 73, 74, 74, 75, 75, 76, 77

Dot Plots

A dot representsone data point for that value.

Mr. Cruz wants to determine the approximate number of pets owned by the students in his class. He has organized the results of his class survey in the graph displayed.

.If x represents 1 student, which data set is represented by the dot plot?(a) 2, 3, 6, 6, 0, 3, 5, 1, 2, 7, 2, 4, 4, 2, 0.

(b) 2, 3, 6, 6, 0, 3, 3, 1, 2, 2, 7, 4, 4, 1, 0.

(c) 2, 4, 6, 6, 0, 3, 4, 1, 2, 0, 3, 7, 5, 2, 3.

(d) 2, 4, 6, 6, 0, 3, 3, 1, 2, 4, 3, 0, 4, 5, 2

The dot plot shows the number of attempts to pass the first level of a video game by a sample of people.

Which set of data is represented by the dot plot?

(a) (3, 4, 2, 4, 3, 5, 2, 6, 3, 1, 6, 6)

(b) (3, 3, 3, 4, 3, 6, 2, 5, 5, 1, 6, 6)

(c) (1, 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6)

(d) (4, 3, 5, 1, 4, 6, 2, 6, 3, 2, 6, 6)

Finding Statistics from Data

Need to be able to find these things from data:1. mean 2. median3. lower quartile4. upper quartile5. range6. interquartile range7. standard deviation

MeanAlso known as the average; represented by x

MedianAlso known as the middle number; if there are an even amount of numbers, take the 2 middle numbers and divide by 2 (just is the median)

21 24 26 | 28 29 36

27 = median

Lower Quartile"median of the bottom half"; Represented by Q1;

Number that has 25% of the data lower than it and 75% of the data that is higher

Upper Quartile"median of the upper half"; represented by Q3

number that has 75% of the data lower than it and 25% of the data that is higher if there are an even amount of numbers

RangeThe maximum value of the data set minus the minimum value

Interquartile RangeThe upper quartile of the data set minus the lower quartile of the data set; represents the middle 50% of the data

Standard DeviationA measure of how spread out the numbers in data are from the mean.

Smaller standard deviation = data is clustered tightly Larger standard deviation = data is spread out

Symbol in calculator = Sx

Finding Statistics of a Data Set

1. Press stat > 1:Edit... > Input data set in L1

2. Press stat > Calc (Right) > 1:1­Var Stats > Go down to calculate > enter

x = meanSx = standard deviationminX = minimumQ1 = lower quartileMed = medianQ3 = upper quartilemaxX = maximum

The calculator does not give Range and Interquartile Range!

To Find the Range:Maximum ­ Minimum

To Find Interquartile Range:Q3 ­ Q1

Find the following from the data given below.

40, 42, 28, 38, 41, 39, 40, 47, 44mean ( ) = 39.9 median = 40

lower quartile = 38.5 upper quartile = 43

range = 19 std deviation = 5.2

interquartile range = 4.5

x

The dot plot shows the number of desserts families ordered with their dinner at a local steakhouse.

According to the dot plot, what is the mean number of desserts ordered per table?

(a) 1.5 desserts (b) 2 desserts

(c) 2.5 desserts (d) 3 desserts

* ** ** * * * * * * ** * * * * * * *0 1 2 3 4 5 6 7

Histograms

Histogram is NOT a bar graph!

It is a graph with bars that show amounts using intervals (intervals must be the same amount)

Finding the total number by adding up the number in each interval will help you be able to answer histogram questions.

2 5 6 8 5 4total = 30

Select the value which best completes the sentence. The coach of a high school basketball team represented the heights of her players, as shown. Of the basketball players on the team, exactly what percent have heights above 180 cm?

(a) 25%

(b) 50%

(c) 75%

(d) 90%

1 2 1 4 3 1912

A group of parents are surveyed to find out how many hours per night their children sleep on average. The histogram shows the results. According to the data, how much longer, on average, do 1­ to 2­year­old children sleep than 6­ to 13­year­old children?

(a) 1 hour (b) 1.5 hours

(c) 2 hours (d) 2.5 hours

Adrian is training to participate in an upcoming 5K race. He runs a mile each night to help build his stamina. Based on the graph of Adrian’s running times, which interval contains the median of the data set?

(a) 10­11

(b) 12­13

(c) 14­15

(d) 16­17

9 14 6 5 3 2 1 = 40

20 data points to the median; since 23 points through the interval 10­11, the median will be in that interval

Box Plots (Box and Whisker)

Box Plots give you 5 pieces of information:

A. Minimum (Min) B. Maximum (Max)

C. Lower Quartile (Q1) D. Upper Quartile (Q3)

E. Median (Med) NO MEAN W/ BOX PLOTS

A BC E D

Susan noted the heights of the rose plants in her garden and represented the heights in a box plot as shown.

Which statement is true?(a) The average height of the rose plants is 75 cm.(b) The minimum height of the rose plants is 65 cm.(c) The heights of 50% of the rose plants are above 75 cm.(d) The heights of 75% of the rose plants are below 65 cm.

(a) The median height of the rose plants is 75 cm.(b) The lower quartile height of the rose plants is 65 cm.(d) The heights of 25% of the rose plants are below 65 cm OR The heights of 75% of the rose plants are below 78 cm.

The box plot shows the game scores of the Central Middle School basketball team over the past two seasons.

Which statement must be true?(a) The interquartile range of the data is 8..

(b) The majority of the data is greater than the median..

(c) The mean of the data is 68..

(d) The upper or third quartile of the data is 70.

(a) The interquartile range of the data is 5.(b) The 50% of the data is greater than the median.(c) The median of the data is 68.

When comparing data, make sure to find all of

the data before comparing!

The height, in inches, of each student in Megan's class is shown. Select the three measures that will be affected if a student who is 77 inches tall joins the class.

w/out w/ 77A. interquartile range 8 8

B. mean 65.3 65.8

C. median 65 65

D. range 18 23

E. standard deviation 4.95 5.32 77

Two of Ms. Lutz' Earth Science classes have 23 students each. Box plots for the recent test scores for these two classes are displayed.

Which statement about the scores is true?(a) The means of the two sets of data are equal.

(b) The lower quartiles of the two sets of data are the same.

(c) More students in third period than in fifth period scored an 87 or above.

(d) Fewer students in third period than in fifth period scored a 70 or below.

(a) The medians of the two sets of data are equal.(b) The upper quartiles of the two sets of data are the same.(c) More students in third period than in fifth period scored a 70 or above.

^^

A basketball coach is comparing the number of points two players scored in each game during the season.

Which statement accurately compares the two players? (a) Player 2 has a higher mean and median than Player 1.

(b) Player 2 has a higher mean, but Player 1 has a higher median.

(c) Player 1 has a higher mean and median than Player 2.

(d) Player 1 has a higher mean, but Player 2 has a higher median.

x = 15.7 med = 18 x = 18 med = 16

A data set consists of the given values. If the value 19 is added to the data set, which two of these statistics will change?

16 18 19 20 23

(a) interquartile range (b) maximum

(c) mean (d) range

(e) minimum (f) median

before w/ 19(a) interquartile range 3.5 2(b) maximum 23 23(c) mean 19.2 19.15(d) range 7 7(e) minimum 16 16(f) median 19 19

OutliersData is officially an outlier if it lies more than 1.5x the interquartile range from the upper or lower quartile.

Q1 = 25 Q3 = 35

Interquartile Range = 10

25 ­ 1.5(10) = 10 35 + 1.5(10) = 50

So data would be an outlier if it is less than 10 or more than 50.

You are at the beach and walk up to a restaurant. The menu with prices is posted next to the door.

Which of these would be considered a mathematical outlier?.

(a) lobster (b) hot dog .

(c) pizza (d) ice creamOne could calculate the actual numbers where the outliers fall, but by looking at the table, answer choice (a) is the only answer that makes sense.

The statistics for a final exam for Mr. Villa’s Spanish class are shown. Which grade would be an outlier for this test?

(a) 52

(b) 60

(c) 98

(d) 100

Mean = 81

Q1 = 75

Median = 83

Q3 = 85

Minimum = 53

Maximum = 99

IQR = 85 ­ 75 = 10

Low Outlier:75 ­ 1.5(10) = 60Any # less than 60 is an outlier.

High Outlier:85 + 1.5(10) = 100Any # greater than 100 is an outlier.

The only # that qualifies as an outlierwould be answer choice (a).

Shape of Data

Data can be...1. Symmetrical (normal)2. Skewed Right 3. Skewed Left

Data that is skewed may have an outlier, which is a point that lies an abnormal distance from the other values in a data set.

It is better to use the mean to represent the center of the data

when there are no outliers..

It is better to use the median to represent the center of the data

when there are outliers.

The table shows the payroll for a small business. What would be the most appropriate measure of center for the salaries?

(a) interquartile range

(b) standard deviation

(c) mean

(d) median

Answer choices (a) and (b) don't make sense because the measure of the center is either the mean or the median. Since there are outliers (Joe/$500,000 and Alec/$450,000) present, then answer choice (d) would be the best answer.

A random survey for two different samples studies the number of hours each individual spends watching television on a selected date.

Sample Y Sample Z 3, 6, 3, 3, 6, 3 3, 4, 3, 2, 5, 18

Which statement is true? (a) The best estimate for the centers of both samples is the mean. .

(b) The best estimate for the centers of both samples is the median. .

(c) The best estimate for the center of Sample Y is the mean, and the best estimate for the center of Sample Z is the median. .

(d) The best estimate for the center of Sample Y is the median, and the best estimate for the center of Sample Z is the mean.

Sample Y does not have any outliers, so the mean would be the best estimate of for the center.Sample Z does have an outlier (18), so the median would be the best estimate of for the center.

Symmetrical/Normal DataMean is approximately equal to median.

In a box plot of data that is symmetrical, the median is located halfway between the lower and upper quartiles.

Data that is Skewed RightThe data has a short left tail and a long right tail. Mean is greater than the median.

In a box plot of data that is skewed right, the median is located closer to the lower quartile.

Data that is Skewed LeftThe data has a long left tail and a short right tail. Mean is less than the median.

In a box plot of data that is skewed left, the median is located closer to the upper quartile.

.

MEAN MEDIANData is skewed RIGHT.

MEAN MEDIANData is skewed LEFT.

Which data set would display a graph that is skewed left? (a) the age at which people get their driver’s license: 17,18,19,20,21

(b) the incomes of five lawyers: $45000, $38000, $125000, $56000, $42000

(c) the number of soccer goals a single player makes: 4,5,7,12,3

(d) the temperature during April in Vermont: 56,35,72,66,68

(a) 17,18,19,20,21 mean = 19 = 19 = median (symmetrical/normal) (b) $45000, $38000, $125000, $56000, $42000 mean = $61,200 > $56,000 = median (right skew)(c) 4,5,7,12,3 mean = 6.2 > 5 = median (right skew)(d) 56,35,72,66,68 mean = 59.4 < 66 = median (left skew)*

85 students took a test in their Algebra class yesterday. If the mean grade was 73% and the median grade was 80%, what is a likely description of the shape of the distribution?.

(a) The distribution is normal.

(b) The distribution is skewed right.

(c) The distribution is skewed left.

(d) The distribution is symmetric.

Since the mean < median, that means that the data is skewed left.

Students were asked how many times they have tripped up the stairs in the past week. Their answers were recorded and plotted in a histogram. What is the best description for theshape of the graph that was created?

(a) mostly symmetric

(b) skewed to the right

(c) skewed to the left

(d) uniform

The data is skewed to the right because the "tail" of the histogram is longer on the right.

Which set of data is skewed most to the right?

(a) 15, 20, 21, 22, 23, 24, 29

(b) 10, 15, 18, 20, 21, 24, 29

(c) 5, 15, 16, 17, 17, 18, 19

(d) 14, 15, 15, 16, 17, 18, 30

(a) 15, 20, 21, 22, 23, 24, 29 mean = 22, median = 22 mean = median (symmetrical/normal)(b) 10, 15, 18, 20, 21, 24, 29 mean = 19.6, median = 20 mean < median (left skew)(c) 5, 15, 16, 17, 17, 18, 19 mean = 15.3, median = 17 mean < median (left skew)(d) 14, 15, 15, 16, 17, 18, 30 mean = 17.9, median = 16 mean > median (right skew)

Simon takes 7 tests in his class. The table shows his scores.

After taking two more tests, the mean of Simon’s test score data increases but the median remains the same. Which could be the scores of the 2 additional tests?

(a) 77 and 78 (b) 58 and 80

(c) 75 and 100 (d) 90 and 94

First, the median is 76. Since we know that the median remains the same, that means that one number is greater than 76 and the other number is less than 76, in order to keep the median = 76. That eliminates answer choices (a) and (d). We know that the mean increases, so that means that we need the lower number to be closer to the 76 and the higher number to be further away, causing the mean to increase. Answer choice (c) is correct because 75 is close to 76 and 100 is further away, while the opposite is true for answer choice (b).