Chapter 5
Categorical A general name for non-numerical data; the data is separated into categories of some kind. Nominal data Categorical data with no implied order. Eg. Eye colours, favourite TV show, favourite music genre. Ordinal data Categorical data with some form of implied order. Eg. A question followed by the respondent selecting a level of satisfaction ie Very good good moderate - bad
Numerical data A general name for data that is numerical in its nature. Discrete data Numerical data where there is a countable number of values possible. Eg. What year were you born?, How many pets do you own? Continuous data Numerical data where there is an infinite number of values possible; often associated with measurement of some kind. The values can be any (real) numbers within a particular range. Eg. How tall are the players on Melbourne Victory squad? (the values recoded varies depending on the accuracy requested or the degree of accuracy possible with out measuring instrument)
We make this distinction between discrete and continuous data because there are different things we can do with each type. However, we often treat continuous data as if it were discrete. This is because we must record the data in some way. With the height example previously, if the question said to record the data to the nearest cm, then we would have discrete possibilities.
A structural method of recording statistical data. If the range of values is too big, group the data into class intervals. No. of siblings Tally marks Frequency 0 III 3 1 IIII 4 2 II 2 Total 9
We can also add in relative frequency and percentage frequency columns to compare values in different frequency tables where the total for each table is different. Relative frequency Divide the frequency by the total number of data values in the table. The relative frequency table should equate to 1. Percentage frequency Multiply the relative percentage by 100. The percentage frequency column should equate to 100.
Although a frequency table is a useful way of recording data, when we group data values we lose information. A stem plot preserves this lost information. Stem plots are usually divided into intervals of 10. Example: 12, 15, 23, 45, 56, 18, 44, 33, 23, 19, 34, 52, 59, 41, 23, 13, 9, 11, 15, 18 Note: We can easily identify the highest and lowest score. If we need to break the stem because of large numbers of leaf values, use a code such as 5 L to represent the values in the lower half of the 50 s and 5 U for the upper half STEM LEAF 0 9 1 1 2 3 5 5 8 8 9 2 3 3 3 3 3 4 4 1 4 5 5 2 6 9
A bar graph is used to display categorical (nominal) information. Bar graphs can easily be constructed from data represented in a frequency table. Gaps between bars to represent we are not dealing with continuous data. Example: Type of vehicle Sedan 16 Ute 7 Truck 8 Stationwagon 6 Motorbike 1 Total 38 Frequency 20 15 10 5 0 Type of Vehicle
A single rectangle is divided into pieces according to the contribution of the category to the whole. An appropriate scale needs to be chosen to make the graph easier to draw. Find the percentage contribution for each category and base the scale on those values. Component Protein Fat Sat Fat Carbs Sugar No. of grams 17 18.2 6.3 34.5 2.6 Note: Number of grams involved = 78.6 No. of grams 0 50 100 Protein Fat Sat Fat Carbs Sugar
When dealing with numerical data we should not really use a bar chart. However, the temptation is strong when we have discrete data Uses strips instead of bars (to make it look different) Graph does not have a title but both axis need labels. Gaps between strips to represent discrete data
When dealing with continuous numerical data we use histograms. Like a bar chart, but the bars are joint together. There are no gaps in the data values so we leave no gaps in the diagram. Leave a half-column gap between the first column and the vertical axis. Note: indicates that that part of the horizontal axis has been left out. Make the intervals the same size.
When we have a frequency table that have already been set up and the intervals are not the same size, information can be misleading. Looking at the histogram, it seems to indicate that the most dangerous age range is 30-<40 years, but this is not correct, because the interval is larger we would expect more deaths. We need a relative measure to make such a decision.
To draw a histogram correctly when we have inconsistent intervals we need to a percentage frequency and a frequency density column. Frequency density Frequency density = percentage frequency class width Remember: Precentage frequency = frequency total number of data values 100
Example: Percentage frequency = 3 100 = 1.0% 299 Frequency density = 1.0 = 0.2 5 Note: add the vertical axis title frequency density
5.3.4: The frequency table gives information about the number of traffic fatalities for females in Victoria during 2002. draw a historgram of the data, taking note of the inconsistent class interval used.
Add the percentage frequency column and a frequency density column. Precentage frequency = frequency 100 total number of data values Frequency density = percentage frequency class width Age range No. of deaths Percentage frequency Frequency density 0-1 (1 / 98) x 100 = 1.0 (1.0 / 5) = 0.2 5-3 (3 / 98) x 100 = 3.1 (3.1 / 8) = 0.3875 13-5 (5 / 98) x 100 = 5.1 (5.1 / 3) = 1.7 16-6 (6 / 98) x 100 = 6.1 (6.1 / 2) = 3.05 18-11 (11 / 98) x 100 = 11.2 (11.2 / 4) = 2.8 22-6 (6 / 98) x 100 = 6.1 (6.1 / 4) = 1.525 26-5 (5 / 98) x 100 = 5.1 (5.1 / 4) = 1.275 30-7 (7 / 98) x 100 = 7.1 (7.1 / 10) = 0.71 40-12 (12 / 98) x 100 = 12.2 (12.2 / 10) = 1.22 50-11 (11 / 98) x 100 = 11.2 (11.2 / 10) = 1.12 60-16 (16 / 98) x 100 = 16.3 (16.3 / 15) = 1.087 75+ 15 (15 / 98) x 100 = 15.3 (15.3 / 15) = 1.02 Σf=98
Age range No. of deaths Percentage frequency Frequency density 0-1 1.0 0.204 5-3 3.1 0.3825 13-5 5.1 1.7 16-6 6.1 3.06 18-11 11.2 2.805 22-6 6.1 1.53 26-5 5.1 1.275 30-7 7.1 0.714 40-12 12.2 1.224 50-11 11.2 1.122 60-16 16.3 1.089 75+ 15 15.3 1.021 Σf=98
When we have a continuous data set recorded in class intervals we can draw what is called a cumulative frequency diagram. This shows the number of data values less than a particular value. Add a column to the frequency table labelled cumulative frequency Add a new first row to emphasis we are finding the number of data values less than the given value. The cumulative frequency column tells us how many values there are less than the right-hand end-point of the interval. Eg.
Example: Class interval (mass, kg) 10 - <20 5 20 - <30 8 30 - <40 16 40 - <50 5 Frequency (number of students) Σf = 34 Class interval (mass, kg) Frequency (number of students) Cumulative frequency <10 0 0 10 - <20 5 5 (0+5) 20 - <30 8 13 (5+8) 30 - <40 16 29 (13+16) 40 - <50 5 34 (29+5) Σf = 34
Once we have added the cumulative frequency column we can now draw the diagram. Cumulative frequency 40 35 30 25 20 15 10 5 0 10 20 30 40 50 Mass (kg)
How to read the diagram What percentage of people are less than 35kg? Find the cumulative frequency: 20 people How many people in data set: 34 20 100 = 58.82% 34 Cumulative frequency 40 30 20 10 0 10 20 30 40 50 59% of people surveyed are less than 35 kg. Mass (kg)
We can use our CAS to find only approximate values for percentiles. These arise from questions such as Under what mass are 70% of the students? of course you have also been given the distribution of the student weight. We have the following information Class interval (mass, kg) Frequency (Number of students) 10 - <20 5 20 - <30 8 30 - <40 16 40 - <50 5 We need to enter the cumulative frequency values as the second column in our list. As a check, the total frequency should be the value in the last row.
Class interval (mass, kg) Frequency (Number of students) Cumulative frequency <10 0 0 10 - <20 5 5 20 - <30 8 13 30 - <40 16 29 40 - <50 5 34 1. Enter the data in the Lists & spreadsheet application. Call the first column x Enter the right-hand boundary values of the class interval Call the second column y Include the point (10,0)
2. Insert a new page,, and choose Data and Statistics. Label the axis 3. Join the dots by pressing Type > XY Line Plot > Plot
4. To estimate the 70 th percentile we need to know 70% of the cumulative frequency. In this case this is 0.7 x 34 = 23.8. we now want to draw in the line y = 23.8. to do this press > Analyse > Plot Function and fill in the dialog box. We can now approximate the x-value of the point of intersection by simply reading from the graph. Therefore, the 70 th percentile is approximately 37
Mean The measure of central tendency found by adding together all of the data values and dividing by the data set sum of the data values number of data values = Σx n = Σxf Σf Median The measure of central tendency that is the (physical) middle value of the data set for odd n, the median is the n 2 + 0.5 th value. for even n, the median is the mean of the n 2 th and n 2 + 1 th values. Mode The measure of central tendency that is the most frequent occurring data value
1. Enter the data in Lists & Spreadsheets using the midpoint values to represent each group. 2. Press > Statistics > Stat Calculations > One- Variable Statistics to bring up this dialogue box.
3. In this case we have only one list of data values as the second list is just the frequencies. So, to OK and press. This brings up another dialogue box which needs to be filled in as shown. (If you had just entered a list of ungrouped data values then you would leave Frequency List as 1 )
4. to OK and press and we get the summary statistics included in our table. Arrow down to highlight the value beside x and read of the value of the mean from the entry line at the bottom. In this case it is 9.4375 5. Continue to arrow down to find the value beside MedianX as this is the median. This tells us the median is 7, but as this is grouped data, we should really say the median occurs in the 5-9 group.
Find the mode, median and mean of the following: Score Frequency 4 15 5 23 6 14 7 23 8 17 9 19 10 20 Mode: look in the frequency column for the highest occurring number Mode = 5 and 7
Median Find Σf Score Frequency 4 15 5 23 6 14 7 23 8 17 9 19 10 20 Score Frequency 4 15 5 23 6 14 7 23 8 17 9 19 10 20 Σf= 131 for odd n, the median is the n 2 + 0.5 th value. Median = 131 2 Median = 7 + 0.5 th value = 66th value = 7
Mean: Add a new column to the frequency table and label it xf (x stands for the data value, f for the frequency), and fill it by multiplying the data values by the frequency for that value. Score Frequency 4 15 5 23 6 14 7 23 8 17 9 19 10 20 Σf= 131 Score Frequency xf 4 15 60 5 23 115 6 14 84 7 23 161 8 17 136 9 19 171 10 20 200 Σf= 131 Σxf = 927 Mean = Σxf Σf = 927 131 = 7.08 (2 dp)
Find the modal class, median class and mean (2 dp) for the following grouped discrete data sets. Score Frequency 10-19 15 20-29 23 30-39 17 40-49 15 50-59 13 60-69 20 70-79 5 Modal class: Look in the frequency column for the highest number; the modal class is the number associated with this. Modal class = 20-29
Median class 108 Find Σf 2 for even n, the median is the mean of the n th and n + 1 th values. 2 2 th value and 108 2 54th and 55 th values + 1 th value 54 th and 55 th values are in the 30-39 interval Median class = 30-39 Score Frequency 10-19 15 20-29 23 30-39 17 40-49 15 50-59 13 60-69 20 70-79 5 Σf= 108
When we deal with grouped data we cannot find a single mode, instead we find the modal class. We can find a specific value for the mean but we need to find the these values that represent each of these class intervals. We use the median of the interval and call this the midpoint of the interval (x m ) We add two new columns to the frequency table x m and x m f
Add the new columns x m and x m f Score Frequency 10-19 15 20-29 23 30-39 17 40-49 15 50-59 13 60-69 20 70-79 5 Score Frequency x m x m f 10-19 15 14.5 217.5 20-29 23 24.5 563.5 30-39 17 34.5 586.5 40-49 15 44.5 667.5 50-59 13 54.5 708.5 60-69 20 64.5 1290 70-79 5 74.5 372.5 Σf= 108 Σx m f = 4406 Find the mean: Mean = Σx mf Σf = 4406 108 = 40.80
For the following data sets draw a cumulative frequency curve to find an estimate for the median. Add a cumulative frequency column to the table. Note that <141 is included as a start to the class. Score Frequency 141-<146 22 146-<151 31 151-<156 21 156-<161 26 161-<166 22 166-<171 24 Score Frequency Cumulative f <141 0 0 141-<146 22 22 146-<151 31 53 151-<156 21 74 156-<161 26 100 161-<166 22 122 166-<171 24 146 Σf= 146
Draw the cumulative frequency curve Score Frequency Cumulative f <141 0 0 141-<146 22 22 146-<151 31 53 151-<156 21 74 156-<161 26 100 161-<166 22 122 166-<171 24 146 Σf= 146 for even n, the median is the mean of the n th and n + 1 th values. 2 2 Median = 146 146 and + 1 = median between 2 2 73rd and 74 th value Median is approximately 156 Draw a line from the median position across to the curve and then draw a line from this point on the curve down to the horizontal axis.
The median divides the data set into two equal pieces. We can extend this idea by dividing the data set into any number of equal-sized pieces. We call these pieces quantiles. Some quantiles have special names. 4 equal pieces = quartiles 10 equal pieces = deciles 100 equal pieces = percentiles
Lower quartile (Q1 or QL) Upper quartile (Q3 or QU) We consider there to be approximately 25% of the data values in each of the quartiles.
Sometimes we want to find how the data spreads out from the central values (mean, median, mode) The range (maxx-minx) is the simplest measure of spread Sensitive to extreme values (outliers) Example: 1, 1, 1, 4, 6, 8, 8, 9, 10, 13, 13, 14 Range = 14 1 = 13 The range represents the difference between the highest score to the lowest score Interquartile range: IQR = Q 3 Q 1 Interquartile range (IQR) = upper quartile (Q 3 ) lower quartile (Q 1 ) The IQR represents the middle 50% of the data set.
5.5.1 Find the interquartile range for the following data set. Q 1 Q 3 2, 3, 5, 8, 9, 11, 15, 16, 18, 25, 36 IQR = Q 3 Q 1 IQR = 18 5 IQR = 13 median
We use quartiles to draw up boxplots. The key features of box plots are show in the diagram. The five values shown are sometimes referred to as the five-figure summary for a data set. The box plot needs a scale line associated with it. The box is the central box representing the middle 50% of the data. The whiskers are the lines that go out to the extreme values.
5.5.3a Find the interquartile range and draw a boxplot for the following data. Find Σf Σf = 24 Identify the median Median = mean of the n th and n + 1 th values. 2 2 Median between 12 th and 13 th value. = 2 Identify Q 1 and Q 3 12 values in the lower and upper half Q 1 is between the 6 th and 7 th value = 1 Q 3 is between the 18 th and 19 th value = 3 Score Frequency 0 3 1 6 2 7 3 4 4 3 5 1 Calculate the IQR IQR = 3 1 = 2
Draw a box plot to represent the data Q 1 = 1 Q 3 = 3 Median = 2
5.5.3c Find the interquartile range and draw a boxplot for the following data. Find Σf Σf = 26 Identify the median Median = mean of the n th and n 2 2 Median between 13 th and 14 th value. = 30.5 Identify Q 1 and Q 3 13 values in the lower and upper half Q 1 is the 7 th value = 15 Q 3 is the20 th value = 42 + 1 th values. Stem Leaf 1 1 2 3 3 4 4 5 8 2 2 3 4 5 3 0 1 1 2 6 6 4 2 2 3 9 5 0 9 6 1 8 Calculate the IQR IQR = 42 15 = 27
Q 1 = 15 Q 3 = 42 Median = 30.5
5.5.5 Draw a box plot for the following data which is displayed in a percentage cumulative frequency graph Draw a line from the 50% mark to estimate for the median Approx median = 18 Draw a line at the 25% mark and 75% mark to estimate Q1 and Q3 Approx Q 1 = 12 Approx Q 3 = 23.5 Calculate the IQR IQR = 23.5 12 = 11.5
Q 1 = 12 Q 3 = 23.5 Median = 18
1. Enter the data the normal way. As we will be drawing a graph you need to name the column. We will use x here. Then get the summary statistics screen by pressing > Statistics > Stat Calculations > One-Variable Statistics. When you get to the dialog box you need to make X1 List x as this is the name you gave the column with the data in it. Leave Frequency List as 1 since we entered each value individually. You will be asked about the x say it is a Variable Reference.
2. Arrow down to get the five-figure summary on the screen and write the values into your exercise book. 3. Insert a new page ( ) and choose Add Data & Statistics. Name the horizontal axis with the name given to column A, in our case x. Now press > Plot Type > Box Plot and our boxplot appears. Move your cursor around to find out info The dot represents an outlier or extreme value. An outlier is a value that lies more than 1.5 x IQR away from the nearer of the upper or lower quartiles.
The standard deviation ( σ ) sigma is a measure of spread relating to how far the data deviates from the mean. I.e. How far from the centre we might expect to still find data. The larger the number, the more spread out the data is. Note: Population: when gather data from the whole group. Ie. The height of year 11 s Every single year 11 is measured. Mean of population: μ (mu) Sample: When we gather data from a select group to represent the population. Ie. The height of year 11 s one class may be measured to represent the whole group. Mean of sample: x (x bar) μ and x are measures of central tendency that tells us where we might expect to find the centre of the data set
We define variance as the mean of the squared deviations from the mean and use the symbol σ 2 (sigma squared) to represent it. Variance = σ 2 = Σ x μ 2 n For grouped intervals σ 2 = Σf x μ 2 Σf sum of the squared deviations from the mean number of values
We squared the deviations to make them positive before adding. To take account of this we now find the positive square root of the variance and call this the standard deviation ( σ ) of the population. So σ = Σf(x μ)2 Σf
There are two standard deviations calculated. sx is the standard deviation if the data represents a sample whereas σx is the standard deviation if the data represents a population. In this case we are using all the members of a team so we would use the σx value. 1. Find the midpoint for each class interval and add them to the table Distance covered (nearest km) Number of runners 21-30 3 25.5 31-40 5 35.5 x 41-50 4 45.5 51-60 2 55.5 61-70 1 65.5 71-80 1 75.5
2. Enter the data and go to the One-Variable Statistics screen and make the entries shown. (it is slightly different to previously since we are not drawing a graph). 3. down to OK and press 4. the mean is x = 43 and the standard deviation we are interested in is σx = 13.92. write these in your book. Note: the calculator has given us the values of the mean ( x), the sum of the x-values (Σx), and the sum of the values 2 (Σx 2 ). Note also that x is used for the mean whereas, strikly speaking, it should be μ, as this was a population. However, there is no difference between the values x and μ.
5.5.8a Find the standard deviation, correct to one decimal place, of the following sample data sets. 3, 3, 4, 6, 2, 1, 3, 4, 6, 7, 5, 3, 2, 1, 7, 9 Enter the data in your CAS using Lists & Spreadsheets Press Menu > Statistics > Stat Calculations > One-Variable Statistics. When prompted choose what you labelled column A sx = 2.3 (1 dp)
Comparative analysis Used when dealing with the analysis of more than one set of data Absolute analysis Used when dealing with the analysis of just one set of data Outliers (Extreme values) A value more than 1.5 x IQR away from the nearer of the upper and lower quartiles
Skewness A term used to describe data sets that are not symmetrical. Boxplots: Negative skew: indicates that the median is closer to the upper quartile Symmetrical: indicates that the median lies in the middle Positive skew: The median is closer to the lower quartile.
Histograms Positive skew: Negative skew: mode < median < mean mean < median < mode Degree of skewness A numerical value to indicate the degree of skewness. The degree of skewness = 3(mean median) standard deviation This value indicates the direction of the skew as well as its size. The larger the value the greater the degree of skewness
Skewed data sets are more likely to contain outliers Use median and IQR as summary statistics Mean and standard deviation are affected by extreme values
5.6.1 Describe the following data set, including a comment about its degree of skewness. The data set represents the score obtained out of 66 on a test. 60, 51, 47, 42, 53, 34, 47, 39, 56, 63, 35, 34, 50, 35, 41, 19, 48, 42, 37, 45, 29 Enter data into CAS, write down the five-figure summary: x (mean) = 43.1905 σx (st dev) = 10.35 minx = 19 Q 1 = 35 Median = 42 Q 3 = 50.5 MaxX = 63
Calculate to determine if there are any outliers in the data set. IQR = 50.5 35 = 15.5 1.5 X IQR = 23.25 No outliers because there is no value that lies further than Q 1 or Q 3 : 50.5 + 23.25 = 73.75 or (maxx = 63) 35 23.5 = 11.5 (MinX = 19) Calculate the degree of skewness 3(mean median) = 3(43.19 42) = 0.34 standard deviation 10.35 Data set contains no outliers, but is slightly positively skewed. This is confirmed by the mean lying closer to the lower quartile and the calculated value for the degree of skewness.
Composite bar charts Some can handle more than two categories
Back-to-back stemplots Can ONLY handle two categories
Comparative boxplots Several related boxplots using the SAME scale Can be used for a number of categories.
Male 54 73 68 50 73 59 75 59 67 68 Female 53 80 74 53 79 63 82 63 73 76 1. In the Lists & Spreadsheets application, name the first column A gender and column B life Enter the male data set in column B. In column A enter male press > Data > Fill down and press. A dashed box will appear. Scroll down column A until the last piece of data and press. 2. Return to column B and under the males life expectancy fill in the female life expectancy. Give all these pieces of data the label female in column A.
To create the box plot, press (to insert a new page) and select Add Data & Statistics. Label the horizontal axis Life and the vertical axis Gender, then press > Plot type > Box plot.
5.6.2. The table below gives the life expectancy for males and females in the 40 most populated countries of the world. Draw comparative boxplots for males and females and discuss your findings. Enter the data into your CAS to produce the boxplots. And five figure summary Both data sets are negatively skewed. Females outscore males on all of the five values. 50% of females values are greater than the value for which less than 25% of the males are greater. There are no outliers. The range of values and IQR are greater for females.
5.6.5. The bar chart below shows the number of drivers killed in accidents that involved the vehicle going off path on straight of going off path on curve for the years 2002 and 2003. compare the two years. In both years many more males were killed than females. In both years the youngest drivers killed were females, these drivers must have been driving illegally as they were under 16. In 2002, three underage male drivers were killed. More males died in 2002 in comparison to 2003 except for the age class 18-21where more males died In 2003 compared to 2002. Female figures are so low that useful comparisons cannot really be made.
5.6.8. The data below represents the height (cm) of the players for two clubs. Draw a back-to-back stemplot and use it to help make comparisons between the teams.
Find the median and IQR of both teams. WB median = 38 and 38 + 1 = between the 2 2 19th and 20 th value = 186 cm WB IQR = 193 181 = 12 Geelong median = 189.5 Geelong IQR = 193 184 = 9 Geelong appears to have the taller team; median height 189.5 cm compared with 186 cm. Spread of height is slightly greater for WB with a larger IQR. WB modal class is 18 L and Geelong s is 19 L reinforcing that Geelong has the taller team.