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lseidman@matcmadison.edu Biotechnology Laboratory Technician Program Course: Basic Biotechnology Laboratory Skills for a Regulated Workplace Lisa Seidman, Ph.D. Ph.D. STATISTICS A BRIEF INTRODUCTION WHY LEARN ABOUT STATISTICS? Statistics provides tools that are used in Quality control Research Measurements Sports lseidman@matcmadison.edu IN THIS COURSE We will use some of these tools Ideas Vocabulary A few calculations lseidman@matcmadison.edu VARIATION There is variation in the natural world People vary Measurements vary Plants vary Weather varies lseidman@matcmadison.edu Variation among organisms is the basis of natural selection and evolution lseidman@matcmadison.edu EXAMPLE 100 people take a drug and 75 of them get better 100 people don’t take the drug but 68 get better without it Did the drug help? lseidman@matcmadison.edu VARIABILITY IS A PROBLEM There is variation in response to the illness There is variation in response to the drug So it’s difficult to figure out if the drug helped lseidman@matcmadison.edu STATISTICS Provides mathematical tools to help arrive at meaningful conclusions in the presence of variability lseidman@matcmadison.edu Might help researchers decide if a drug is helpful or not This is a more advanced application of statistics than we will get into lseidman@matcmadison.edu DESCRIPTIVE STATISTICS Chapter 16 in your textbook Descriptive statistics is one area within statistics lseidman@matcmadison.edu DESCRIPTIVE STATISTICS Provides tools to DESCRIBE, organize and interpret variability in our observations of the natural world lseidman@matcmadison.edu DEFINITIONS Population: Entire group of events, objects, results, or individuals, all of whom share some unifying characteristic lseidman@matcmadison.edu POPULATIONS Examples: All of a person’s red blood cells All the enzyme molecules in a test tube All the college students in the U.S. lseidman@matcmadison.edu SAMPLE Sample: Portion of the whole population that represents the whole population lseidman@matcmadison.edu Example: It is virtually impossible to measure the level of hemoglobin in every cell of a patient Rather, take a sample of the patient’s blood and measure the hemoglobin level lseidman@matcmadison.edu MORE ABOUT SAMPLES Representative sample: sample that truly represents the variability in the population -good sample lseidman@matcmadison.edu TWO VOCABULARY WORDS A sample is random if all members of the population have an equal chance of being drawn A sample is independent if the choice of one member does not influence the choice of another Samples need to be taken randomly and independently in order to be representative lseidman@matcmadison.edu SAMPLING How we take a sample is critical and often complex If sample is not taken correctly, it will not be representative lseidman@matcmadison.edu EXAMPLE How would you sample a field of corn? lseidman@matcmadison.edu VARIABLES Variables: Characteristics of a population (or a sample) that can be observed or measured Called variables because they can vary among individuals lseidman@matcmadison.edu VARIABLES Examples: Blood hemoglobin levels Activity of enzymes Test scores of students lseidman@matcmadison.edu A population or sample can have many variables that can be studied Example Same population of six year old children can be studied for Height Shoe size Reading level Etc. lseidman@matcmadison.edu DATA Data: Observations of a variable (singular is datum) May or may not be numerical Examples: Heights of all the children in a sample (numerical) Lengths of insects (numerical) Pictures of mouse kidney cells (not numerical) lseidman@matcmadison.edu ALWAYS UNCERTAINTY Even if you take a sample correctly, there is uncertainty when you use a sample to represent the whole population Various samples from the same population are unlikely to be identical So, need to be careful about drawing conclusions about a population, based on a sample – there is always some uncertainty lseidman@matcmadison.edu SAMPLE SIZE If a sample is drawn correctly, then, the larger the sample, the more likely it is to accurately reflect the entire population If it is not done correctly, then a bigger sample may not be any better How does this apply to the corn field? lseidman@matcmadison.edu INFERENTIAL STATISTICS Another branch of statistics Won’t talk about it much Deals with tools to handle the uncertainty of using a sample to represent a population lseidman@matcmadison.edu EXAMPLE PROBLEM In a quality control setting, 15 vials of product from a batch are tested. What is the sample? What is the population? In an experiment, the effect of a carcinogenic compound was tested on 2000 lab rats. What is the sample? What is the population? lseidman@matcmadison.edu A clinical study of a new drug was tested on fifty patients. What is the sample? What is the population? lseidman@matcmadison.edu ANSWERS 15 vials, the sample, were tested for QC. The population is all the vials in the batch. The sample is the rats that were tested. The population is probably all lab rats. The sample is the 50 patients tested in the trial. The population is all patients with the same condition. lseidman@matcmadison.edu EXAMPLE PROBLEM An advertisement says that 2 out of 3 doctors recommend Brand X. What is the sample? What is the population? Is the sample representative? Does this statement ensure that Brand X is better than competitors? lseidman@matcmadison.edu ANSWER Many abuses of statistics relate to poor sampling. The population of interest is all doctors. No way to know what the sample is. The sample could have included only relatives of employees at Brand X headquarters, or only doctors in a certain area. Therefore the statement does not ensure that the majority of doctors recommend Brand X. It certainly does not ensure that Brand X is best. lseidman@matcmadison.edu DESCRIBING DATA SETS Draw a sample from a population Measure values for a particular variable Result is a data set lseidman@matcmadison.edu DATA SETS Individuals vary, therefore the data set has variation Data without organization is like letters that aren’t arranged into words lseidman@matcmadison.edu Numerical data can be arranged in ways that are meaningful – or that are confusing or deceptive lseidman@matcmadison.edu DESCRIPTIVE STATISTICS Provides tools to organize, summarize, and describe data in meaningful ways Example: Exam scores for a class is the data set What is the variable of interest? Can summarize with the class “average”, what does this tell you? lseidman@matcmadison.edu A measure that describes a data set, such as the average, is sometimes called a “statistic” Average gives information about the center of the data lseidman@matcmadison.edu MEDIAN AND MODE Two other statistics that give information about the center of a set of data Median is the middle value Mode is most frequent value lseidman@matcmadison.edu MEASURES OF CENTRAL TENDENCY Measures that describe the center of a data set are called: Measures of Central Tendency Mean, median, and the mode lseidman@matcmadison.edu HYPOTHETICAL DATA SET 2 5 6 7 8 3 9 3 10 4 7 4 6 11 9 Simplest way to organize them is to put in order: 2 3 3 4 4 5 6 6 7 7 8 9 9 10 11 By inspection they center around 6 or 7 lseidman@matcmadison.edu MEAN Mean is basically the same as the average Add all the numbers together and divide by number of values 2 3 3 4 4 5 6 6 7 7 8 9 9 10 11 What is the mean for this data set? lseidman@matcmadison.edu NOMENCLATURE Mean = 6.3 = read “X bar” The observations are called X1, X2, etc. There are 15 observations in this example, so the last one is X15 Mean = Xi n Where n = number of values lseidman@matcmadison.edu EXAMPLE Data set 2 3 3 4 5 6 7 8 9 What is the mode? What is the median? lseidman@matcmadison.edu MEAN OF A POPULATION VERSUS THE MEAN OF A SAMPLE Statisticians distinguish between the mean of a sample and the mean of a population The sample mean is The population mean is μ It is rare to know the population mean, so the sample mean is used to represent it lseidman@matcmadison.edu DISPERSION Data sets A and B both have the same average: A 4 5 5 5 6 6 B 1 2 4 7 8 9 But are not the same: A is more clumped around the center of the central value B is more dispersed, or spread out lseidman@matcmadison.edu MEASURES OF DISPERSION Measures of central tendency do not describe how dispersed a data set is Measures of dispersion do; they describe how much the values in a data set vary from one another lseidman@matcmadison.edu MEASURES OF DISPERSION Common measures of dispersion are: Range Variance Standard deviation Coefficient of variation lseidman@matcmadison.edu CALCULATIONS OF DISPERSION Measures of dispersion, like measures of central tendency, are calculated Range is the difference between the lowest and highest values in a data set lseidman@matcmadison.edu Example: 2 3 3 4 4 5 6 6 7 7 8 9 9 10 11 Range: 11-2 = 9 or, 2 to 11 Range is not particularly informative because it is based only on two values from the data set lseidman@matcmadison.edu CALCULATING VARIANCE AND STANDARD DEVIATION Variance and standard deviation measure of the average amount by which each observation varies from the mean Example: 4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm Data set, lengths of 8 insects lseidman@matcmadison.edu 4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm The mean is 7 cm How much do they vary from one another? Intuitively might see how much each point varies from the mean This is called the deviation lseidman@matcmadison.edu CALCULATION OF DEVIATIONS FROM MEAN 4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm Value-Mean in cm Deviation (4-7) (5-7) (6-7) (7-7) (7-7) (7-7) (9-7) (11-7) -3 -2 -1 0 0 0 +2 +4 lseidman@matcmadison.edu Value-Mean Deviation (in cm) (4-7) (5-7) (6-7) (7-7) (7-7) (7-7) (9-7) (11-7) -3 -2 -1 0 0 0 +2 +4 Sum of deviations = lseidman@matcmadison.edu 0 Sum of the deviations from the mean is always zero Therefore, cannot use the average deviation Therefore, mathematicians decided to square each deviation so they will get positive numbers lseidman@matcmadison.edu Value-Mean Deviation SquaredDeviation (in cm) (4-7) (5-7) (6-7) (7-7) (7-7) (7-7) (9-7) (11-7) -3 -2 -1 0 0 0 +2 +4 9 cm2 4 cm2 1 cm2 0 0 0 4 cm2 16 cm2 total squared deviation = sum of squares = lseidman@matcmadison.edu 34 cm2 VARIANCE Total squared deviation (sum of squares) divided by the number of measurements: 34 cm2 = 4.25 cm2 8 lseidman@matcmadison.edu STANDARD DEVIATION Square root of the variance: 4.25 cm2 = 2.06 cm Note that the SD has the same units as the data Note also that the larger the variance and SD, the more dispersed are the data lseidman@matcmadison.edu VARIANCE AND SD OF POPULATION VS SAMPLE Statisticians distinguish between the mean and SD of a population and a sample The variance of a population is called sigma squared, σ2 Variance of a sample is S2 lseidman@matcmadison.edu The standard deviation of a population is called sigma, σ Standard deviation of a sample is S or SD lseidman@matcmadison.edu STANDARD DEVIATION OF A SAMPLE (Xi - )2 n -1 lseidman@matcmadison.edu EXAMPLE PROBLEM A biotechnology company sells cultures of E. coli. The bacteria are grown in batches that are freeze dried and packaged into vials. Each vial is expected to have 200 mg of bacteria. A QC technician tests a sample of vials from each batch and reports the mean weight and SD. lseidman@matcmadison.edu Batch Q-21 has a mean weight of 200 mg and a SD of 12 mg. Batch P-34 has a mean weight of 200 mg and as SD of 4 mg. Which lot appears to have been packaged in a more controlled fashion? lseidman@matcmadison.edu ANSWER The SD can be interpreted as an indication of consistency. The SD of the weights of Batch P-34 is lower than of Batch Q-21. Therefore, the weights for vials for Batch P-34 are less dispersed than those for Batch Q-21 and Batch P-34 appears to have been better controlled. lseidman@matcmadison.edu FREQUENCY DISTRIBUTIONS So far, talked about calculations to describe data sets Now talk about graphical methods lseidman@matcmadison.edu TABLE 5 THE WEIGHTS OF 175 FIELD MICE (in grams) 19 21 19 20 19 20 22 22 23 21 20 22 25 25 24 26 22 21 24 20 24 20 22 22 21 20 22 21 22 26 20 22 21 23 21 21 21 21 23 22 21 22 21 22 20 20 20 21 23 22 25 21 21 22 23 20 22 19 23 22 21 23 23 21 23 21 24 22 23 25 22 23 22 24 24 25 21 22 22 19 22 24 19 24 22 23 20 21 22 24 25 21 25 21 23 23 23 21 19 19 24 21 23 20 20 20 24 26 20 23 lseidman@matcmadison.edu 19 24 22 22 22 24 20 21 18 23 21 22 21 23 28 21 26 21 21 21 21 22 27 21 19 27 24 19 23 25 20 22 24 24 22 22 20 23 22 23 22 22 25 20 25 17 22 23 21 22 20 23 24 20 20 23 22 23 20 20 22 24 23 22 FREQUENCY DISTRIBUTION TABLE OF THE WEIGHTS OF FIELD MICE Weight (g) Frequency 17 18 19 20 21 22 23 24 25 26 27 1 1 11 25 34 40 27 19 10 4 2 28 1 lseidman@matcmadison.edu FREQUENCY TABLE Tells us that most mice have weights in the middle of the range, a few are lighter or heavier The word distribution refers to a pattern of variation for a given variable lseidman@matcmadison.edu It is important to be aware of patterns, or distributions, that emerge when data are organized by frequency The frequency distribution can be illustrated as a frequency histogram lseidman@matcmadison.edu FREQUENCY HISTOGRAM X axis is units of measurement, in this example, weight in grams Y axis is the frequency of a particular value For example, 11 mice weighed 19 g The values for these 11 mice are illustrated as a bar lseidman@matcmadison.edu Note that when the mouse data were collected, a mouse recorded as 19 grams actually weighed between 18.5 g and 19.4 g. Therefore the bar spans an interval of 1 gram lseidman@matcmadison.edu FIRST FOUR BARS F R E Q U E N C Y 17 18 19 20 WEIGHTS IN GRAMS lseidman@matcmadison.edu CONSTRUCTING A FREQUENCY HISTOGRAM Divide the range of the data into intervals It is simplest to make each interval (class) the same width No set rule as to how many intervals to have For example, length data might be 1-9 cm, 10-19 cm, 20-29 cm and so on lseidman@matcmadison.edu Count the number of observations that are in each interval Make a frequency table with each interval and the frequency of values in that interval Label the axes of a graph with the intervals on the X axis and the frequency on the Y axis lseidman@matcmadison.edu Draw in bars where the height of a bar corresponds to the frequency of the value Center the bars above the midpoint of the class interval For example, if the interval is 0-9 cm, then the bar should be centered at 4.5 cm lseidman@matcmadison.edu NORMAL FREQUENCY DISTRIBUTION If weights of very many lab mice were measured, would likely have a frequency distribution that looks like a bell shape, also called the “normal distribution” lseidman@matcmadison.edu NORMAL DISTRIBUTION F R E Q U E N C Y WEIGHT lseidman@matcmadison.edu NORMAL DISTRIBTION Very important Examples: Heights of humans Measure same thing over and over, measurements will have this distribution lseidman@matcmadison.edu CALCULATIONS AND GRAPHICAL METHODS Related The center of the peak of a normal curve is the mean, the median and the mode Values are evenly spread out on either side of that high point lseidman@matcmadison.edu The width of the normal curve is related to the SD The more dispersed the data, the higher the SD and the wider the normal curve Exact relationship is in text, not go into it this semester lseidman@matcmadison.edu EXAMPLE PROBLEM A technician customarily performs a certain assay. The results of 8 typical assays are: 32.0 mg 28.9 mg 23.4 mg 30.7 mg 23.6 mg 21.5 mg 29.8 mg 27.4 mg a. If the technician obtains a value of 18.1 mg, should he be concerned? Base your answer on estimation. b. Perform statistical calculations to see if the answer if out of the range of two SDs. lseidman@matcmadison.edu ANSWER The average appears to be in the midtwenties and hovers around + 5. Therefore, 18.1 mg appears a bit low. Mean = 27.16 mg, SD = 3.87 mg. The mean – 2SD is 19.4 mg, so 18.1 mg appears to be outside the range and should be investigated lseidman@matcmadison.edu lseidman@matcmadison.edu
