#### Transcript brief_intro_descriptive_statistics - Bio-Link

[email protected] 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 [email protected] IN THIS COURSE We will use some of these tools Ideas Vocabulary A few calculations [email protected] VARIATION There is variation in the natural world People vary Measurements vary Plants vary Weather varies [email protected] Variation among organisms is the basis of natural selection and evolution [email protected] 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? [email protected] 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 [email protected] STATISTICS Provides mathematical tools to help arrive at meaningful conclusions in the presence of variability [email protected] Might help researchers decide if a drug is helpful or not This is a more advanced application of statistics than we will get into [email protected] DESCRIPTIVE STATISTICS Chapter 16 in your textbook Descriptive statistics is one area within statistics [email protected] DESCRIPTIVE STATISTICS Provides tools to DESCRIBE, organize and interpret variability in our observations of the natural world [email protected] DEFINITIONS Population: Entire group of events, objects, results, or individuals, all of whom share some unifying characteristic [email protected] 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. [email protected] SAMPLE Sample: Portion of the whole population that represents the whole population [email protected] 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 [email protected] MORE ABOUT SAMPLES Representative sample: sample that truly represents the variability in the population -good sample [email protected] 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 [email protected] SAMPLING How we take a sample is critical and often complex If sample is not taken correctly, it will not be representative [email protected] EXAMPLE How would you sample a field of corn? [email protected] VARIABLES Variables: Characteristics of a population (or a sample) that can be observed or measured Called variables because they can vary among individuals [email protected] VARIABLES Examples: Blood hemoglobin levels Activity of enzymes Test scores of students [email protected] 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. [email protected] 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) [email protected] 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 [email protected] 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? [email protected] 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 [email protected] 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? [email protected] A clinical study of a new drug was tested on fifty patients. What is the sample? What is the population? [email protected] 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. [email protected] 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? [email protected] 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. [email protected] DESCRIBING DATA SETS Draw a sample from a population Measure values for a particular variable Result is a data set [email protected] DATA SETS Individuals vary, therefore the data set has variation Data without organization is like letters that aren’t arranged into words [email protected] Numerical data can be arranged in ways that are meaningful – or that are confusing or deceptive [email protected] 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? [email protected] 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 [email protected] 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 [email protected] MEASURES OF CENTRAL TENDENCY Measures that describe the center of a data set are called: Measures of Central Tendency Mean, median, and the mode [email protected] 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 [email protected] 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? [email protected] 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 [email protected] EXAMPLE Data set 2 3 3 4 5 6 7 8 9 What is the mode? What is the median? [email protected] 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 [email protected] 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 [email protected] 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 [email protected] MEASURES OF DISPERSION Common measures of dispersion are: Range Variance Standard deviation Coefficient of variation [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 = [email protected] 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 [email protected] 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 = [email protected] 34 cm2 VARIANCE Total squared deviation (sum of squares) divided by the number of measurements: 34 cm2 = 4.25 cm2 8 [email protected] 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 [email protected] 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 [email protected] The standard deviation of a population is called sigma, σ Standard deviation of a sample is S or SD [email protected] STANDARD DEVIATION OF A SAMPLE (Xi - )2 n -1 [email protected] 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. [email protected] 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? [email protected] 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. [email protected] FREQUENCY DISTRIBUTIONS So far, talked about calculations to describe data sets Now talk about graphical methods [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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 [email protected] FIRST FOUR BARS F R E Q U E N C Y 17 18 19 20 WEIGHTS IN GRAMS [email protected] 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 [email protected] 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 [email protected] 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 [email protected] 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” [email protected] NORMAL DISTRIBUTION F R E Q U E N C Y WEIGHT [email protected] NORMAL DISTRIBTION Very important Examples: Heights of humans Measure same thing over and over, measurements will have this distribution [email protected] 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 [email protected] 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 [email protected] 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. [email protected] 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 [email protected] [email protected]