QUANTITATIVE DATA. UNIVARIATE DATA data for one variable

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1 QUANTITATIVE DATA Recall that quantitative (numeric) data values are numbers where data take numerical values for which it is sensible to find averages, such as height, hourly pay, and pulse rates. UNIVARIATE DATA data for one variable GRAPH The distributions of quantitative univariate data can be graphed using the following: Dotplot Stemplot Histogram Boxplot Ogive Dotplot dots represent observations (each can represent multiple observations (provide key); used with few categories, small data set, small range; column heights show relative frequency 1) Must have a title. 2) Draw using only a horizontal axis. 3) Each value in the range of the data must be represented. 4) This type of graph should generally only be used for integer values. EXAMPLE: Stemplot allows for the display of original data values and provides an easy way to sort values; number of leaves is equal to the number of total data entries 1) Must have a title and a key (3 2 = 32, 4 1 = 4.1, etc.). 2) There are 2 n stems that are optimal, where n is the size of the data set (min 5 stems). When stems are split, an equal number of leaves must be available for each stem; therefore, you can only split into 2s or 5s. It may be necessary to split the stems if the data set has a small range or has many leaves for each stem. 3) Do not include unused stems at top or bottom, but do not skip stems in the middle. EXAMPLE: 14 4 = 144

2 The stemplot above is created without splitting the stems. There is only one of each number value in the Stem column (5-15), where each stem has 10 possible leaf values, 0-9. However, sometimes, in order to meet the optimal number of stems, stems must be split into 2s or 5s, resulting in duplicates of number values in the Stem column: Stems split into 2s Stems split into 5s etc Each stem has 5 possible leaf values, Each stem has 2 possible leaf values, and , 2 3, 4 5, 6 7, and 8 9. EXAMPLE: Determine the final number of stems to be used. (a) n = 80, range (b) n = 49, range EXAMPLE: Make a stemplot using the given data set. (a) Chapter 1 Test Grades, n = 27, range

3 Histogram most commonly used graph for representing quantitative variables, large data sets Frequency tables provide a basis for constructing histograms. Frequency Distribution a table that lists data values along with their corresponding frequencies (counts) or relative frequencies (percents) 1) Must have title, two columns (class, frequency). 2) List the quantitative data values under Class. Quantitative data values are usually grouped into intervals (a range of values) with a minimum of five classes: a. Find the number of k classes using 2 k n (where n is the number of data points and k is the smallest integer that makes true statement). b. Find class width using r +1 (highest value) - (lowest value) + 1 k (desired number of classes) (where r is the range [high-low] and k is the number of classes). 3) Show corresponding counts under Frequency or percents under Relative Frequency. a. Use tally marks to represent data values for each class interval. b. Add tallies to find frequencies, or divide frequencies by n to find relative frequencies. EXAMPLE: Construct a frequency table from the reported number of hours volunteered by doctors. Number of Hours Doctors Volunteered Class Frequency Number of Volunteer Hours Reported Frequencies can be presented as percentages cumulative (accruing) counts cumulative percentages RELATIVE FREQ. DISTRIB. CUMULATIVE FREQ. DISTRIB. CUM. REL. FREQ. DISTRIB. Number of Volunteer Hours Number of Volunteer Hours Number of Volunteer Hours Class Relative Frequency Class Cumulative Frequency Class Cum. Rel. Frequency % % % % % % % % % % Sometimes the relative frequencies do not sum to exactly 100%... this is due to round-off error. (a.k.a. OGIVE)

4 Histogram(cont d) most used graph for representing quantitative variables, large data sets 1) Begin with frequency table: a. Find the number of k classes (minimum 5) using 2 k n (where n is the number of data points and k is the smallest integer that makes true statement). b. Find class width using r +1 k 2) Must have a title and both axes labeled (include units if necessary) 3) Bars should be uniform with no gaps (unless class interval is empty, containing zero). (where r is the range [high-low] and k is the number of classes). EXAMPLE: Community Service Hours Submitted by Students Number of employees Number of hours volunteered *Notice that the absence of bars (between 0-5 and 20-25) indicates that there are zero observations in those intervals; do not omit empty intervals with histograms. EXAMPLE: Construct a histogram for doctor volunteer hours. EXAMPLE: On graph paper, construct a relative frequency histogram for the number of daily bank withdrawals:

5 Modified Boxplot graph showing the center, spread, and distribution of the data, as well as any outliers; useful for skewed data sets that contain outliers; graph of the Five-Number Summary 1) Must have title and horizontal axis, labeled and numbered in regular increments 2) Construct using medians, maximum and minimum values, and outliers. a. Box is constructed with 1 st amd 3 rd quartiles at ends and 2 nd quartile (median) at center. b. Whiskers extend from the box to the minimum and maximum non-outlier values. c. Always plot outliers separately. 3) Interpret a boxplot by first focusing on the middle 50% in the box to determine shape. Ogive cumulative histogram; can be used to determine how many data values lie above or below a particular value in a data set 1) Using cumulative frequency table 2) Must have a title and both axes labeled (include units if necessary) 3) Connect the points with straight lines EXAMPLE: EXAMPLE: Find the at the 40 th percentile.

6 DESCRIPTION When describing data sets, provide a description of the distribution in context of the data/variables that are being measured, making note of the overall shape, outliers/deviations, the value at the center, and the range of values, using as a guide SOCS: Shape, Outliers, Center, Spread. SOCS: Shape, Outliers, Center, Spread Be sure to provide information about the following important characteristics in the description of a distribution, always in context. Shape the nature or shape of the graphical distribution as symmetric, skewed, etc. Mean Median Mode SYMMETRIC DISTRIBUTION LEFT-SKEWED BELL-SHAPED RIGHT-SKEWED When the number of values is approximately the same for all variables of a data set, the shape of the distribution is described as uniform (consider the distribution of very many die rolls). (pic) Mentally remove outliers before determining shape of the distribution as symmetric, skewed, uniform, etc., noting any outliers. Identify the location where the majority of the data values fall; make note of the location of peaks, modes: unimodal, bimodal, multimodal; comment on any gaps/clusters. Be sure to describe the shape in context, in terms of the variable being measured, including units of measure where necessary, providing plausible explanations. (store hours, internet use at work) EXAMPLE: Describe the shape of the distribution of doctor volunteer hours. The shape of the distribution of the number of hours volunteered by doctors is right-skewed and unimodal with the most volunteered at 2 hours. Outliers extreme sample values that lie very far from the majority of others Always use modified boxplots that show outliers graphed separately. Outliers are data values that lie beyond 1½ IQRs above the 3 rd quartile and 1½ IQRs below the 1 st quartile. Outliers must be verified using the following formulas: Low Outlier LO < Q 1 1.5(IQR) High Outlier HO > Q (IQR) IQR (Inter-Quartile Range) the range of the middle 50% of the data set; the difference between the 1 st and 3 rd quartiles, Q 3 Q 1.

7 Be sure to describe the outliers in context, in terms of the variable being measured, providing plausible explanation for such an extreme value. EXAMPLE: Comment on any outliers in the distribution of doctor volunteer hours. The is an outlier in the distribution of the number of hours volunteered by doctors at 24 hours. LO < Q 1 1.5(Q 3 Q 1 ) HO > Q (Q 3 Q 1 ) LO < 3 1.5(11 3) HO > (11 3) LO < 3 12 HO > LO < -9 HO > 23 Center the median or the mean The median is the middle value of an ordered set of data points and is not affected by outliers: the median is resistant to the effects of outliers and, therefore, is used with skewed distributions or those with outliers. The mean is the average value x and is affected by outliers: the mean is not resistant to n the effects of outliers and, therefore, is used with relatively symmetric distributions. The mean and median are the same/close in symmetric distributions. Be sure to describe measures of center in context, in terms of the variable being measured. EXAMPLE: Describe the center of the distribution of doctor volunteer hours. The median number of hours volunteered by doctors is 5, and the mean is 7.5 hours. Spread a measure of the amount of the variability among data values The range is sometimes limited as a description of spread because it is affected by outliers (the range is non-resistant ), so the IQR is a useful measure of spread with skewed distributions or those with outliers. The smaller the spread, the less variation among data values The larger the spread, the more variation among data values Less variability is good for businesses and consumers, as it demonstrates consistency in production and products. Standard deviation is another measure of variability for symmetric distributions. Be sure to describe measures of spread in context, in terms of the variable being measured. EXAMPLE: Comment on the spread of the distribution of doctor volunteer hours. The range of number of hours volunteered by doctors is 23 hours (24 1), with an IQR of 8 hours (11 3).

8 Numerical Summary The numerical summary will depend on the nature of the distribution. A numerical summary of center and spread uses either mean and standard deviation for approximately symmetric distributions or median and IQR for skewed distributions (or those with outliers). For relatively symmetric distributions, use mean and standard deviation (both are nonresistant to outliers). Mean x the sum of all data values divided by number of values, Σx ; affected by n outliers, non-resistant Standard deviation s the typical distance the data points are from the mean; a measure of dispersion of the values in a data set; affected by outliers, non-resistant; larger s indicates more variation; s is zero when all the data values are the same Σ(x x )2 s = n 1 EXAMPLE: Find the standard deviation for the following heights (in inches) From calculator: 65.8 s = For skewed distributions, use median and IQR (and 5-Number Summary) because both are resistant to outliers. Be sure to include that explanation in your description ( resistant ). Median med the middle value of ordered data set values; not affected by outliers, resistant IQR interquartile range the range of the middle 50% of data values; Q 3 Q 1 ; not affected by outliers, resistant Five-Number Summary 1. the minimum value 2. the first quartile, Q 1 3. the median (Q 2 ) 4. the third quartile, Q 3 5. the maximum value The graph of the 5-number summary is a boxplot: Median Minimum Q 1 Q 2 Q 3 Maximum Outlier

9 EXAMPLE: On graph paper, construct a relative frequency histogram and a stemplot for local city temperatures (Fahrenheit), and provide a description/numerical summary of the distribution (PRACTICE: Triola, p. 761, January s Actual Low Temperatures) Linear Transformations Sometimes, we will to convert units of measure of the data, as with conversions from Fahrenheit to Celsius, for example. Such a linear transformation will change the original values into new values by: y = a + bx. / Adding a constant a shifts all data values up or down by a as well as with measures of center (median and mean) and position (quartiles and percentiles), but affects no changes to measures of spread (range, IQR, standard deviation) or shape. / Multiplying by a constant b changes the size of the units of measurement and multiplies both the measures of center, position, and spread by b. COMPARISON Often, we will want to compare two or more data sets, such as with different types of treatments and with different groups of test subjects. Use comparative language to identify, explain, and support the bigger, better data set of the two. Also remember to consider quartiles and outliers. The following graphs can be used: Parallel Dotplot Back-to-back stemplot Double histogram Parallel boxplots Back-to-back stemplot compare multiple groups using same stems (using same process as above, with the lowest value of both and the highest value of either) EXAMPLE:

10 Parallel boxplot compare two groups using same x-axis; used to compare quartiles and variability EXAMPLE: Runs Scored per Game for the National and American League Teams Number of Runs Scored EXAMPLE: Provide a description/summary/comparison of driver reaction times of the two groups. Driver Reaction Times Overall, it appears that Group 1 had faster reaction times than Group 2. Group 1 has lower/better minimum, Q1, Q3, and maximum reaction time values. Although Group 2 has a lower median reaction time (1.8 sec) than the median reaction time of Group 1 (1.9 sec), it is only by 0.1 sec. Group 1 also shows less variability with a range of 1.4 sec, compared to Group 2 s range of 2.4 sec, and an IQR of 0.7 sec, compared to Group 2 s IQR of 0.8 sec. (PRACTICE: Triola, p. 748, 749, Compare cholesterol levels of males and females, with a graph, description, and numerical summary.)