Statistics -- The methods or procedures that researchers apply in an attempt to understand data. Data -- Information in numerical form that represents a certain characteristic Statistics are a collection of theory and methods applied for the purpose of understanding data. Presented by Brent Daigle, Ph.D. (ABD)
The summation operator is often used to simplify statistical formulas. The Greek capital letter (sigma) denotes the summation operator. The Summation Operator Presented by Brent Daigle, Ph.D. (ABD)
For example, suppose we have the following five numbers: X 1 = 3 X 2 =7 X 3 =4 X 4 = 2 X 5 =8 The subscripts 1 through 5 on the X’s simply mean that they stand for different numbers. If we want to sum the five numbers, we proceed as follows: 5 i = 1 X i = X 1 + X 2 + X 3 + X 4 + X 5 = 3 + 7 + 4 + 2 + 8 = 24 Presented by Brent Daigle, Ph.D. (ABD)
means that the summation begins with the first number and concludes with the N th. Often the notations above and below the are omitted. When they are, means summation from the first through the N th or last number. X i (called “cap X , subscript i ” or just “ X sub i ”) is the general symbol for the number. The notation under the Indicates the first number in the summation (in this case, X 1 = 3), and the number above the Indicates how far the summation continues (in this case, through X 5 = 8 ). In general, the notation N i = 1 X i Presented by Brent Daigle, Ph.D. (ABD)
N i = 1 CX i = C N i = 1 X i Rule 1: Applying the summation operator, , to the products Resulting form multiplying a set a numbers by a constant (unchanging value) is equal to multiplying the constant by the sum of the numbers. Symbolically, if C is a constant, RULE 1 Presented by Brent Daigle, Ph.D. (ABD)
Example For example, let 2 be the constant, and let the numbers be X 1 = 1, X 2 = 3, X 3 = 4, and X 4 = 7. Then we find 4 i = 1 (2)Xi = 2(1) + 2(3) + 2(4) + 2(7) = 2 + 6 + 8 + 14 = 30 AND 2 4 i = 1 X i = 2(1 + 3 + 4 + 7) = 2(15) = 30 Presented by Brent Daigle, Ph.D. (ABD)
Rule 2: Applying the summation operator, , to a series of constant scores is the same as taking the product of N times the constant score. Symbolically, N i = 1 C = NC Suppose X 1 = 4, X 2 = 4, and X 3 = 4; that is, each of three scores is equal to the constant 4. Then 3 i = 1 C = 4 + 4 + 4 = 12 AND NC = 3(4) = 12 Presented by Brent Daigle, Ph.D. (ABD)
Rule Applying the summation operator, , to the algebraic Sum of two (or more) scores for a single individual and then summing these sums over the N individuals is the same as summing each of the two (or more) scores separately over the N individuals and then summing the scores. Symbolically, N i = 1 (X i + Y i ) = N i = 1 X i + N i = 1 Y i Presented by Brent Daigle, Ph.D. (ABD)
Suppose we have four individuals with the following X and Y scores: Continued on next slide Presented by Brent Daigle, Ph.D. (ABD) Individual X Y 1 2 7 2 5 9 3 3 6 4 1 5
We find that 4 i = 1 ( X i + Y i ) = ( X 1 + Y 1 ) + ( X 2 + Y 2 ) + ( X 3 + Y 3 ) + ( X 4 + Y 4 ) = (2 + 7) + (5 + 9) + (3 + 6) + (1 + 5) = 9 + 14 + 9 + 6 = 38 and 4 4 i = 1 i = 1 Y i = ( X 1 + X 2 + X 3 + X 4 ) + ( Y 1 + Y 2 + Y 3 + Y 4 ) = (2 + 5 + 3 + 1) + (7 + 9 + 6 + 5) = 11 + 27 = 38 Presented by Brent Daigle, Ph.D. (ABD)
The summation operator, , is one of the most widely used symbols in Statistics. It indicates the summing of the numbers that immediately Follow it in an expression Presented by Brent Daigle, Ph.D. (ABD)
Two special indicators of arithmetic operations appear often in statistics. These are exponent and square root symbols. Squaring a number, or multiplying a number by itself, is a common calculation in statistics. For example, to find the square of Y, we multiply Y times Y : (Y)(Y)=(5)(5)=25. Symbolically, the square is denoted Y 2 , or Y to the second power. The number 2 is called the exponent , the power to which the number is raised. If Y is raised to the fourth power, Y 4 , we have Y 4 = (5) (5) (5) (5) = 625. Ernest Hemingway’s Swimming Pool Presented by Brent Daigle, Ph.D. (ABD)
Finding the square root of a number is also a common calculation. The Square Root of a number is a real number such that, when you multiply the square root by itself, the product is the original number. The symbol over the number indicates that the square root is to be taken. For example, = 6 because 6 2 = (6)(6) = 36. The square root of a number can also be indicated by the fractional exponent ½; that is , = (36) 1/2 = 6 36 36 Presented by Brent Daigle, Ph.D. (ABD)
Absolute Value The absolute value of a number can be considered as the distance between 0 and that number on the real number line. Remember that distance is always a positive quantity (or zero). The distance in the diagram above from +4 to 0 is 4 units and the distance from -3 to 0 is 3 units. These units are never negative values. We write this as 4 = 4 and -3 = 3 Presented by Brent Daigle, Ph.D. (ABD)
Exponents or Square Root Presented by Brent Daigle, Ph.D. (ABD)
After every arithmetic operation that yields a fraction, carry the decimal to three places and round to two places more than there were in the original data. Presented by Brent Daigle, Ph.D. (ABD)
Nominal Data Nominal Data : Labels attached to something. E.g., Male/female, yes/no. Simply puts into a category. Presented by Brent Daigle, Ph.D. (ABD) Discrete vs. continuous A full ratio scale can be modeled on a number line. A number line is continuous which means that there are no gaps between the numbers. However, empirically we can only measure things with a degree of inaccuracy. That means there are gaps in our measure, and this we refer to as discrete . Continuous measures: Time Height Weight Discrete measures: The number of cars in a car park The results of an examination (or homework) The cost of anything
Ordinal Rank – These stars are put in order according to their brightness Presented by Brent Daigle, Ph.D. (ABD) Ordinal Scale This is where data can be recognized as being in some order . For example, a collection of names might be ordered alphabetically . Or a list of entrants in an exam might be ordered by their marks . The order does not mean that items can be added because the gap between the items is unspecified.
Interval Scale – In relation to itself … No ABSOLUTE ZERO Presented by Brent Daigle, Ph.D. (ABD) Interval Scale This is where the gaps between whole numbers on the scale are equal. This permits the arithmetic operations of addition and subtraction. An example of this kind of scale is temperature. 20° C is not twice as hot as 10° C because 0° is not an absolute zero. It is simply the amount of heat beyond which water turns from solid to liquid.
Presented by Brent Daigle, Ph.D. (ABD) Ratio Scale A ratio scale permits full arithmetic operation. Measuring the time something takes is an example of using a ratio scale. If a train journey takes 2 hr and 35 min, then this is half as long as a journey which takes 5 hr and 10 min. We need to be very careful about the scale which is being used, and it is not uncommon for people to do arithmetic on scales which are simply ordinal - marks for assignments at school for example!
A measuring system having the property that ratios of the numbers assigned accurately reflect ratios of the magnitudes of the objects being measured. This means that a behavior assigned a number of, say, 10, must have twice the magnitude of a behavior assigned a number of 5. Effectively, this requires that there be a true zero, a score that means the behavior is absent. Ratio scales in psychology are almost always physical scales employed to capture behavior, such as the use of response latency to measure task difficulty. Typical “psychological” dimensions, eg, intelligence or attitude, do not have a “zero” Presented by Brent Daigle, Ph.D. (ABD)
Presented by Brent Daigle, Ph.D. (ABD) A research project which makes use of statistical analysis can be broken down in the following way: 1. Clarify the question that you are asking. Identify what would count as indicators for the question you are looking at. 2. Identify a suitable source for data and arrange a system of collection. This might include surveys, interviews, questionnaires, observations, counting and so on. 3. Sort , classify and tabulate the data. 4. Carry out the statistical analysis . 5. Present your statistics for further interpretation .
Constant - In mathematics and the mathematical sciences, a constant is a fixed , but possibly unspecified, value. This is in contrast to a variable, which is not fixed. An object whose value cannot be changed. Variable - Observable characteristics that vary among individuals. Dependent Variable - a variable that is not under the experimenter's control -- the data. It is the variable that is observed and measured in response to the independent variable. A variable that receives stimulus and measured for the effect the treatment has had upon it. Independent Variable - In the EXPERIMENTAL METHOD, the INDEPENDENT VARIABLE is the factor that the researcher believes will cause something to happen. It is the presumed "cause" in the cause-and-effect relationship that the experimental method seeks to demonstrate. (eg In a study attempting to demonstrate that a particular drug is effective in relieving anxiety, the drug is the independent variable.) Despite the name, the independent variable does not truly vary independently; it is typically controlled by the person conducting the research. Presented by Brent Daigle, Ph.D. (ABD)
Population ~ The total number of individuals or objects being analyzed or evaluated. Sample ~ used in survey sampling to describe the group (of people, homes, etc.) selected for interviewing in the survey. The sample, by definition, has to be fully representative of the wider group (the population) from which it is drawn. In statistics, a group of observations selected from a statistical population by a set procedure. Samples may be selected at random or systematically. The sample is taken in an attempt to estimate the population. See random sample. Parameter ~ A "constant" or numerical description of some property of a population (which may be real or imaginary). Cf. statistic." A variable, measurable property whose value is a determinant of the characteristics of a system; eg, temperature, pressure, and density are parameters of the atmosphere. Statistic ~ A statistic is an estimate of a population parameter that is inferred from a sample. Usually the parameter is unknowable, but an estimate of the parameter is possible. Statistics are numerical properties of samples. For example, there is a mean income for the people living in a particular country. That is the parameter. The mean income obtained by sampling a part of the population is a statistic. See parameter. Presented by Brent Daigle, Ph.D. (ABD)
Presented by Brent Daigle, Ph.D. (ABD) Descriptive Statistics ~ A statistic used to describe a set of cases upon which observations were made. FOR EXAMPLE, the average age of a class in high school calculated by using all members of that class. A statistic that numerically summarizes an important characteristic of a set of numbers. Inferential Statistics ~ The next step above descriptive statistics, inferential statistics uses significance tests and other measures to make inferences are made about the data - such as using a sample to make estimate about a population. Statistics used to determine whether changes in a dependent variable are caused by an independent variable. Inferential statistics are used to draw logical conclusions about a population from a sample. There are two main methods used in inferential statistics: estimation and hypothesis testing. In estimation, the sample is used to estimate a parameter and a confidence interval about the estimate is constructed. In the most common use of hypothesis testing, a null hypothesis is put forward and it is determined whether the data are strong enough to reject it.
THE END ! Presented by Brent Daigle, Ph.D. (ABD)