Grouped vs. Ungrouped Data Grouped Data – Data that has been organized into groups (into a frequency distribution). If you see a table similar to the one below, you will know that you are dealing with grouped data: Class 0–5 6 – 10 11 – 15 16 – 20
Frequency 4 5 12 7
The frequency of a class is the number of numbers in that class. For example, there must have been four numbers between 0 and 5.
Ungrouped Data – Data that has not been organized into groups. Ungrouped data looks like a big ol’ list of numbers.
How to Group Data On your exam, you may have to construct a frequency distribution. Constructing a frequency distribution is the same thing as grouping data. The first step in grouping data is deciding how large of a class interval to use. (Class interval = Class size) There are 2 formulas for determining the appropriate class interval. You must be able to choose which one would be appropriate for any given problem.
1. Class interval
=
Use when the problem states the number of classes to be used.
2. Class interval =
Use when the problem does not state the number of classes to be used.
**Don’t forget to always round up to the nearest whole number when dealing with class interval.**
Populations vs. Samples Population – A collection of all possible individuals, objects, or measurements (Mason 7). Please note that a population does not have to be huge in size. It is all a matter of how the objects in the group are defined. For example, if we wanted to compute the average GPA of the population of students taking BA254 at GRCC in the summer of 2002, our population would only include approximately 75 people. On the other hand, if we wanted to find the average GPA of all students who have ever taken BA254 at GRCC, we would be dealing with a much larger number of students. This is because of how each population was defined. In the first example, we gave our population a definition that severely limited the number of GPA’s that would be included. The second population was defined a little more broadly, and, therefore, more students would fall under this definition. Sample – A portion or part of the population. (Mason 7)
Measures of Central Tendency Central Tendency – Where the numbers tend to cluster. Where most of the numbers are at. A single value that summarizes a set of data by locating the center of the data (Mason 65). 4 Basic Measures of Central Tendency: 1. The Mean = The Average = The Arithmetic Mean 2. The Median = The Middle (of the road) 50% of the data fall above the median, and 50% fall below the median. 3. The Mode = The Most The mode is the number(s) that appear(s) the most out of a given set of data. A data set can have more than one mode value. 4. Geometric Mean = G.M. The geometric mean is a measure of central tendency that is particularly useful for averaging percentages (%). It is sometimes considered to be a more “conservative” average. This is because of the fact that the GM is always less than or equal to the Arithmetic Mean.
Measures of Dispersion What is Dispersion? To measure the dispersion of a set of numbers means to measure how spread out the numbers in the set are. Dispersion = Spread. Some Measures of Dispersion: 1. Range = Highest Value – Lowest value The range is the simplest measure of dispersion; it only takes into account the highest and lowest values. 2. Mean Deviation = M.D. M.D. involves all the values of a set of data into its calculation. 2
3. Variance = s or ó
2
s2 = Sample variance ó 2 = Population variance 4. Standard Deviation = s or ó s = Sample standard deviation ó = Population standard deviation **Please note that the variance is equal to the standard deviation squared. This also means that the standard deviation is the square root of the variance. Therefore, if we know either the standard deviation or the variance, we can always determine the other very quickly. For example, if standard deviation = 5, then variance = 52 = 25 Or If Variance = 100, then standard deviation =
= 10.
Formulas for Ungrouped Data **Please remember how to distinguish between Grouped and Ungrouped Data (On previous pages!)** **Also, please note that you will have to differentiate between populations and samples when dealing with ungrouped data.** MEAN = ARITHMETIC MEAN = AVERAGE = the sample mean ì
=
= the population mean =
**The mean for ungrouped data will be calculated in the exact same way, regardless of whether you are dealing with a sample or a population. Later in this class, you will need to differentiate between ì and X. It is, therefore, a good idea to become aware of the definitions of ì and X now.** MODE = MOST Whichever number or numbers that appear the most within a list (set) of data. RANGE = A SIMPLE MEASURE OF SPREAD Range = Highest value – Lowest value VARIANCE = s2 or ó 2
**When dealing with Ungrouped data, you must determine whether you are dealing with a sample or a population when computing the standard deviation or the variance. Be careful!!! (If it is a sample, you will likely see the word “sample” in the problem. If it is a population, you may or may not see the word “population” in the problem. In other words, if you do not see the word “sample”, assume that you are dealing with a population!) s2 = Sample Variance =
ó 2 = Population Variance
=
=
STANDARD DEVIATION = s or ó s = Sample Stand. Dev.
=
ó = Population Stand. Dev. =
=
GEOMETRIC MEAN = G.M. The GM is useful when averaging percentages and also when looking for an average percentage increase over time (taking into account the effects of compounding). There are 2 G.M. formulas on your formula sheet. You need to know when and where to use each! GEOMETRIC MEAN FORMULA #1 This formula is appropriate where you have a big ol’ list of numbers (ungrouped data). a G.M. = a a n = number of numbers X1, X2 = each individual value GEOMETRIC MEAN #2 This formula is appropriate where you have a beginning as well as an ending value. This formula, therefore, also involves time, and it is likely to involve years and/or dates. G. M. =
n n n
n = number of time periods (e.g. number of years)
MEAN DEVIATION = M.D. M.D. =
a a a a
**The straight up and down brackets indicate absolute value. Please remember that this means to add up all of the differences without paying attention to whether they are negative or positive! Mean deviation is another measure of spread (others include range, std. deviation, variance). MEDIAN (UNGROUPED) Steps: 1. Locate the median. Location of Median = This formula tells us the rank of the median in terms of smallest to largest. 2. Put the numbers in order from smallest to largest. 3. Determine the value of the median by applying the number you got from the location of median formula to the ascending list of numbers that you just created. For example, if the location of the median is equal to 7, then the median is the 7th number in order of smallest to largest. Using another example, if the location of the median is equal to 7.5, then the median is halfway (.5) between the 7th and the 8th numbers in order from smallest to largest.
Q1, Q3, and Quartile Deviation Q1 = First Quartile Q1 is the value that has 25% of the data set below it, and 75% above it. Computing Q1 and Q3 is very similar to the way the median is determined. In fact, the median could also be called Q2. Steps: 1. Locate Q1. Location of Q1= 2. Put the numbers in order from smallest to largest. 3. Determine the value of Q1 by applying the number you got from the location of Q1 formula to the ascending list of numbers that you just created. Value of Q1 = (Higher number of the two – lower number of the two) * the percentage to the right of the decimal place from you result to the location formula + the lower of the two numbers Q3 = Third Quartile Q3 is the value that has 25% of the data set below it, and 75% above it. Computing Q1 and Q3 is very similar to the way the median is determined. In fact, the median could also be called Q2. Steps: 1. Locate Q3. Location of Q3= 2. Put the numbers in order from smallest to largest. 3. Determine the value of Q3 by applying the number you got from the location of Q3 formula to the ascending list of numbers that you just created. Value of Q3 = (Higher number of the two – lower number of the two) * the percentage to the right of the decimal place from you result to the location formula + the lower of the two numbers
Formulas for Grouped Data Problems **Please remember how to distinguish between Grouped and Ungrouped Data (On previous pages)** If you are presented with a problem consisting of Grouped Data, you will need to use some formulas and techniques which are unique to Grouped Data Problems. **Also, please note that you will not have to differentiate between populations and samples when dealing with grouped data.** **I will use the following example to illustrate all of the following calculations relating to grouped data: Class 0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25
Frequency = f 2 7 12 6 3
MEAN = ARITHMETIC MEAN = AVERAGE MEAN for Grouped Data = Class 0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25
fX = f multiplied by X 5 52.5 150 105 + 67.5 380 = • fx = the sum of fx **Take note of the difference between how we figure out n in this GROUPED data problem, as opposed to how we would solve for n in an UNGROUPED data problem(Here we add the frequencies, in Ungrouped we would just count the number of numbers). In either case, n is the number of numbers!
=
Frequency = f 2 7 12 6 + 3 30 = n
=
=
Midpoint = X 2.5 7.5 12.5 17.5 22.5
380/30 = 12.667
STANDARD DEVIATION (GROUPED) = s STANDARD DEVIATION for Grouped Data = s = Class 0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25
f Midpoint = X 2 2.5 7 7.5 12 12.5 6 17.5 + 3 22.5 30 = n
fX = f multiplied by X X2 5 6.25 52.5 56.25 150 156.25 105 306.25 + 67.5 506.25 380 = • fX
fX2 12.5 393.75 1875 1837.5 + 1518.75 5637.5 = • fX2
s= s = 5.331 VARIANCE (GROUPED) = s2 s = 5.331, so s2 = 5.3312 = 28.420 MEDIAN Steps for Grouped Data Median: 1. Locate which class contains the median by using the following formula: Location of median =
= 15.5
This formula tells you what “place” the median would be in if the numbers were placed in order from smallest to largest! In this example, the median is located halfway between the 15th and the 16th numbers. 2. Now figure out which class would contain the median. (Remember the median in this example is between the 15th and the 16th numbers in order from smallest to largest.) Class 0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25
Frequency = f 2 7 12 6 3
Cumulative Frequency 2 9 21 27 30
The median is somewhere between 10 and 15. We know this because we can look at the Cumulative Frequency and see that the 15th and the 16th numbers are in this class! (In fact, the 10th through the 21st numbers were in this class) 3. Do the Math! **This is one of the trickier formulas for this exam! Please take close note of what the following variables stand for in this particular problem: Median = L = Lower limit of the class that contains the median = 10 n = Number of numbers = 30 CF = Number of numbers before the class containing the median = 9 f = number of numbers in the class containing the median = 12 i = class interval (size) Median = MODE (GROUPED) = MOST The mode for a grouped data problem is the midpoint of the class with the highest frequency (f). The class 10 – 15 has the highest frequency (12), so the mode is the midpoint of this class. MODE = 12.5 **Please take close note that the formulas used for computing the mean, the median, and the standard deviation for GROUPED data, all contain lowercase “f” in them !!!! I would recommend that you use this fact to help remember which formulas are appropriate for the grouped data problems.
RANGE (GROUPED) Range (Grouped) = Upper Limit of Highest Class – Lower Limit of Lowest Class In our example, 0-5 is the lowest class and 0 is the lower limit of this class. Also, 20-25 is the highest class, and 25 is the upper limit of this class. Range(Grouped) = 25 – 0 = 25
Uses of Standard Deviation Chebyshev’s Theorem = where k = the number of standard deviations from the mean. Many years ago, Russian mathematician P.L. Chebyshev developed a theorem that allows us to determine, for any set of data, the minimum proportion of values that lie within a specified number of standard deviations from the mean. Basically, Chebyshev’s Theorem states that at least 1- (1/k2) of the data must fall between the points + ks and - ks. So, one could say that at least 75% of the data in any given set will fall within 2 standard deviations of the mean. Proof 1-(1/k2) = 1 – (1/22) = 1 – ¼ = .75 Further, if it was known that the mean of the same set of data was equal to 20 and the standard deviation was equal to 5, one could conclude that at least 75% would fall between 10 and 30. Proof We proved that 75% fall within 2 standard deviations of the mean when we did that calculation a couple of seconds ago. Now we need to figure out how far two standard deviations from the mean is in our particular example! We need to compute + + ks and + ks = 20 + 2*5 = 20 +10 = 30 - ks = 20 – 2*5 = 20 – 10 = 10
- ks:
Empirical Rule (For the “normal” distribution)
***(Grouped or Ungrouped)***
First of all, the normal distribution is a set of numbers where the 3 main measurements of central tendency (the mean, median, and mode) are all approximately equal. Graphically, a normal distribution is bell-shaped and symmetrical. For the normal distribution, we can be more precise as to what percent of the data will fall within a stated “distance” from the mean. MEMORIZE THE FOLLOWING PERCENTAGES AND THE ASSOCIATED INTERVALS ± 1S ± 2S ± 3S X ± M.D.
68% 95% 99.7% 57.5%
In addition to memorizing the percentages dealing with the normal curve above, make sure to remember that 57.5% of the data for any set of numbers will fall between – M.D. and +M.D.
Coefficient of Variation = C.V. a CV = a a
“A direct comparison of 2 or more measures of dispersion (such as the standard deviation for a distribution of annual incomes compared to the standard deviation for a distribution of the number of days absent for the same group of employees) is impossible. It is impossible because the two values are measured in different units. The absenteeism would be measured in number of days, while the incomes would be measured in dollar bills($). In order to make a meaningful comparison of the 2 standard deviations, we need to convert them to a relative value.” (Mason 113) The measure that is often used to make these types of comparisons is the Coefficient of Variation.
Coefficient of Skewness = Sk Sk = The coefficient of skewness is another measure of where the data in a distribution are located. This formula might indicate that an extreme value (high or low) has affected the mean in a significant way. The coefficient of skewness generally ranges from –3 to +3. If Sk<0, then distribution is negatively skewed. If Sk>0, then distribution is positively skewed. If Sk is approximately = to 0, then the data is normally distributed.
