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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
IN THIS COURSE
We will use some of these tools Ideas Vocabulary A few calculations
VARIATION
There is variation in the natural world People vary Measurements vary Plants vary Weather varies
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?
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
STATISTICS
Provides mathematical tools to help arrive at meaningful conclusions in the presence of variability
Might help researchers decide if a drug is helpful or not
This is a more advanced application of statistics than we will get into
DESCRIPTIVE STATISTICS
Chapter 16 in your textbook Descriptive statistics is one area within
statistics
DESCRIPTIVE STATISTICS
Provides tools to DESCRIBE, organize and interpret variability in our observations of the natural world
DEFINITIONS
Population: Entire group of events, objects, results, or
individuals, all of whom share some unifying characteristic
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.
SAMPLE
Sample: Portion of the whole population that represents the whole population
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
MORE ABOUT SAMPLES
Representative sample: sample that truly represents the variability in the population -- good sample
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
SAMPLING
How we take a sample is critical and often complex
If sample is not taken correctly, it will not be representative
VARIABLES
Variables: Characteristics of a population (or a sample) that
can be observed or measured Called variables because they can vary among
individuals
VARIABLES
Examples: Blood hemoglobin levels Activity of enzymes Test scores of students
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.
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)
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
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?
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
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?
A clinical study of a new drug was tested on fifty patients. What is the sample? What is the population?
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.
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?
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.
DESCRIBING DATA SETS
Draw a sample from a population Measure values for a particular variable Result is a data set
DATA SETS
Individuals vary, therefore the data set has variation
Data without organization is like letters that aren’t arranged into words
Numerical data can be arranged in ways that are meaningful – or that are confusing or deceptive
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?
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
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
MEASURES OF CENTRAL TENDENCY Measures that describe the center of a data
set are called: Measures of Central Tendency Mean, median, and the mode
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
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?
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
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
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
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
MEASURES OF DISPERSION
Common measures of dispersion are: Range Variance Standard deviation Coefficient of variation
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
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
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
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
CALCULATION OF DEVIATIONS FROM MEAN
4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm
Value-Mean Deviationin cm(4-7) - 3(5-7) - 2(6-7) - 1(7-7) 0(7-7) 0(7-7) 0(9-7) +2(11-7) +4
Value-Mean Deviation(in cm)
(4-7) - 3(5-7) - 2(6-7) - 1(7-7) 0(7-7) 0(7-7) 0(9-7) +2(11-7) +4
Sum of deviations = 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
Value-Mean Deviation SquaredDeviation(in cm)
(4-7) - 3 9 cm2
(5-7) - 2 4 cm2
(6-7) - 1 1 cm2
(7-7) 0 0 (7-7) 0 0 (7-7) 0 0(9-7) +2 4 cm2
(11-7) +4 16 cm2
total squared deviation = sum of squares = 34 cm2
VARIANCE
Total squared deviation (sum of squares) divided by the number of measurements:
34 cm2 = 4.25 cm2
8
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
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
The standard deviation of a population is called sigma, σ
Standard deviation of a sample is S or SD
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.
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?
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.
FREQUENCY DISTRIBUTIONS
So far, talked about calculations to describe data sets
Now talk about graphical methods
TABLE 5THE WEIGHTS OF 175 FIELD MICE
(in grams)19 22 20 24 22 19 27 2021 22 20 22 24 24 21 2519 21 20 23 25 22 19 1720 20 21 25 21 22 27 2219 22 23 22 25 22 24 2320 21 22 23 21 24 19 2122 22 25 22 23 20 23 2222 26 21 24 23 21 25 2023 20 21 24 23 18 20 2321 22 22 25 21 23 22 2420 21 23 21 19 21 24 2022 23 20 22 19 22 24 2025 21 22 22 24 21 22 2325 21 19 19 21 23 22 2224 21 23 22 23 28 20 2326 21 22 24 20 21 23 2022 23 21 19 20 26 22 2021 22 23 24 20 21 23 2224 21 23 22 24 21 22 2420 22 21 23 26 21 22 2324 21 23 20 20 21 25 2220 22 21 21 23 22
FREQUENCY DISTRIBUTION TABLE OF THE WEIGHTS OF FIELD MICE
Weight Frequency (g) 17 1
18 119 1120 2521 3422 4023 2724 1925 1026 4
27 2
28 1
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
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
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
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
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
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
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
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”
NORMAL DISTRIBTION
Very important Examples:
Heights of humans Measure same thing over and over,
measurements will have this distribution
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
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
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 mga. 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.
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
