Psych Statistics-Chap 4
Showing means and standard deviations in frequency distribution graphs:
(a) A population distribution with a mean of μ=80 and a standard deviation of σ=8 (b) A sample with a mean of M=16 and a standard deviation of S=2 [Note: In Figure 4.6(a), we show the standard deviation as a line to the right of the mean. You should realize that we could have drawn the line pointing to the left, or we could have drawn two lines (or arrows), with one pointing to the right and one pointing to the left, as in Figure 4.6(b). In each case, the goal is to show the standard distance from the mean.]
What is the result of this correction/adjustment?
-The result of the adjustment is that sample variance provides a much more accurate representation of the population variance -Specifically, dividing by n − 1 produces a sample variance that provides an unbiased estimate of the corresponding population variance
Population distributions of adult male heights and weights:
-The two distributions differ in terms of central tendency: the mean height is 70 inches (5 feet, 10 inches) and the mean weight is 170 pounds -The distributions differ in terms of VARIABILITY -For example, most heights are clustered close together, within 5 or 6 inches of the mean -Weights, on the other hand, are spread over a much wider range
How can the definitional formula for SS be awkward to use?
-When the mean is not a whole number, the deviations all contain decimals or fractions, and the calculations become difficult -Calculations with decimal values introduce the opportunity for rounding error, which can make the result less accurate
What are 3 measures of variability?
1. The range 2. Standard deviation 3. The variance
A good measure of variability serves what 2 purposes?
1. Variability describes the distribution of scores. Specifically, it tells whether the scores are clustered close together or are spread out over a large distance. Usually, variability is defined in terms of distance. It tells how much distance to expect between one score and another, or how much distance to expect between an individual score and the mean (ex. we know that the heights for most adult males are clustered close together, within 5 or 6 inches of the average. Although more extreme heights exist, they are relatively rare.) 2. Variability measures how well an individual score (or group of scores) represents the entire distribution. This aspect of variability is very important for inferential statistics, in which relatively small samples are used to answer questions about populations (ex. suppose that you selected a sample of one adult male to represent the entire population. Because most men have heights that are within a few inches of the population average (the distances are small), there is a very good chance that you would select someone whose height is within 6 inches of the population mean. For men's weights, on the other hand, there are relatively large differences from one individual to another. For example, it would not be unusual to select an individual whose weight differs from the population average by more than 30 pounds) Thus, variability provides information about how much error to expect if you are using a sample to represent a population.
If every individual is classified as either 1, 2, or 3 then there are three measurement categories and the range is:
3 points
By this definition, scores having values from 1 to 5 cover a range of:
4 points
According to this definition, scores having values from 1 to 5 cover a range of:
5.5-0.5=5 points
As a rule of thumb, roughly __% of the scores in a distribution are located within a distance of one standard deviation from the mean, and almost all of the scores (roughly __%) are within two standard deviations of the mean
70%, 95%
Remember that the formulas for sample variance and standard deviation were constructed so that the sample variability provides:
A good estimate of population variability-for this reason, the sample variance is often called estimated population variance, and the sample standard deviation is called estimated population standard deviation
Because the standard deviation measures distance from the mean, it is represented by:
A horizontal line or an arrow drawn from the mean outward for a distance equal to the standard deviation and labeled with a σ or an s
In statistics, our goal is to measure the amount of variability for:
A particular set of scores, a distribution
However, calculating the value of M places:
A restriction on the variability of the scores in the sample
The goal is to find a clear difference between two means that would demonstrate:
A significant, meaningful pattern in the results (variability plays an important role in determining whether a clear pattern exists)
For rough sketches, you can identify the mean with:
A vertical line in the middle of the distribution
What happens to the standard deviation if you add a constant to each score?
Adding a constant to each score does not change the standard deviation
Occasionally a set of scores is transformed by:
Adding a constant to each score or by multiplying each score by a constant value
Behavioural scientists must deal with the variability that comes from studying people and animals. People are not all the same; they have different:
Attitudes, opinions, talents, IQs, and personalities
Although no individual sample is likely to have a mean and variance exactly equal to the population values, both the sample mean and the sample variance, on _____, do provide accurate estimates of the corresponding population values
Average
Why is the range considered to be a crude and unreliable measure of variability?
Because the range does not consider all the scores in the distribution, it often does not give an accurate description of the variability for the entire distribution (therefore, in most situations, it does not matter which definition you use to determine the range)
If the largest distance from the mean is 5 points and the smallest distance is 1 point, the standard deviation should be:
Between 1 and 5
The fact that a sample tends to be less variable than its population means that sample variability gives a _____ estimate of population variability
Biased
In reporting the results of a study, the researcher often provides descriptive information for both:
Central tendency and variability
The scale of measurement helps:
Complete the picture of the entire distribution and helps to relate each individual score to the rest of the group
Standard deviation measures variability by:
Considering the distance between each score and the mean
Give an example of how variability can be viewed as measuring consistency:
Consistency in performance from trial to trial is viewed as a skill. For example, the ability to hit a target time after time is an indication of skilled performance in many sports.
Give an example of how variability can be viewed as measuring diversity:
Corporations, colleges, and government agencies often make attempts to increase the diversity of their students or employees. Once again, they are referring to the differences from one individual to the next.
This definition works well for variables with:
Precisely defined upper and lower boundaries
The sample is said to have n − 1:
Degrees of freedom (how many are "free" to vary)
Although we can calculate the average value for any of these variables, it is equally important to:
Describe the variability
Standard deviation is primarily a _____ measure; it describes how variable, or how spread out, the scores are in a distribution
Descriptive
If you have a small group of scores and the mean is a whole number, then the _____ formula is fine; otherwise the _____ formula is usually easier to use
Directional, computational
Defining the range as the number of measurement categories also works for:
Discrete variables that are measured with numerical scores (ex. if you are measuring the number of children in a family and the data produce values from 0 to 4, then there are five measurement categories (0, 1, 2, 3, and 4) and the range is 5 points)
In frequency distribution graphs, we identify the position of the mean by:
Drawing a vertical line and labeling it with μ or M
In the context of inferential statistics, the variance that exists in a set of sample data is often classified as:
Error variance
Except for minor changes in notation, calculating the sum of squared deviations, SS, is:
Exactly the same for a sample as it is for a population
Note that the sample formula has:
Exactly the same structure as the population formula
Note that the two formulas produce:
Exactly the same value for SS (although the formulas look different, they are in fact equivalent)
In general, low variability means that:
Existing patterns can be seen clearly
High variability tends to obscure any patterns in the data. This general fact is perhaps even more convincing when the data are presented in a:
Graph
Like the mean (μ), variance and standard deviation are parameters of a population and are identified by:
Greek letters
The standard deviation line should extend approximately:
Halfway from the mean to the most extreme score
Using any of these definitions, the range is probably the most obvious way to describe:
How spread out the scores are—simply find the distance between the maximum and the minimum scores
How can the logic and the equations that follow be easier to remember?
If you remember that our goal is to measure the standard, or typical, distance from the mean
Give an example of how variability can be viewed as measuring predictability:
If your morning commute to work or school always takes between 15 and 17 minutes, then your commuting time is very predictable and you do not need to leave home 60 minutes early just to be sure that you arrive on time
Although there are several different ways to picture the data, one simple technique is to:
Imagine (or sketch) a histogram in which each score is represented by a box in the graph
Although the concept of a deviation score and the calculation SS are almost exactly the same for samples and populations, the minor differences in notation are:
Important
Graphs showing the results from two experiments:
In Experiment A, the variability is small and it is easy to see the 5-point mean difference between the two treatments. In Experiment B, however, the 5-point mean difference between treatments is obscured by the large variability.
This fact makes the sample mean and sample variance extremely valuable for use as:
Inferential statistics
Specifically, the mean, the standard deviation, and the variance should be used only with numerical scores from:
Interval or ordinal scales of measurement?
The dependent variables in psychology research are often numerical values obtained from measurements on:
Interval or ratio scales
As the error variance increases:
It becomes more difficult to see any systematic differences or patterns that might exist in the data
Fortunately, the bias in sample variability is consistent and predictable, which means:
It can be corrected (ex. For example, if the speedometer in your car consistently shows speeds that are 5 mph slower than you are actually going, it does not mean that the speedometer is useless. It simply means that you must make an adjustment to the speedometer reading to get an accurate speed-in the same way we will make an adjustment in the calculation of sample variance)
Notice that a few extreme scores in the population tend to make the population variability relatively:
Large
The purpose of the adjustment is to:
Make the resulting value for sample variance an accurate and unbiased representative of the population variance
Standard deviation uses the _____ of the distribution as a reference point
Mean
With numerical scores the most common descriptive statistics are the _____ (central tendency) and the _____ (variability), which are usually reported together
Mean, standard deviation
One commonly used definition of the range simply:
Measures the difference between the largest score and the smallest score
Standard deviation describes variability by:
Measuring distance from the mean
What happens to the standard deviation if you multiply each score by a constant?
Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant
Notice that the sample formulas divide by n − 1 unlike the population formulas, which divide by:
N (This is the adjustment that is necessary to correct for the bias in sample variability)
Does this correction mean that each individual sample variance will be exactly equal to its population variance?
No (some sample variances will overestimate the population value and some will underestimate it-but the average of all the sample variances will produce an accurate estimate of the population variance = unbiased statistic)
Because the standard deviation and variance are defined in terms of distance from the mean, these measures of variability are used only with:
Numerical scores that are obtained from measurements on an interval or a ratio scale
High variability tends to:
Obscure any patterns that might exist
When you have only a sample to work with, the variance and standard deviation for the sample provide the best possible estimates of the:
Population variability
The calculations of variance and standard deviation for a sample follow the same steps that were used to find:
Population variance and standard deviation (First, calculate the sum of squared deviations (SS). Second, calculate the variance. Third, find the square root of the variance, which is the standard deviation.)
Variability can also be viewed as measuring:
Predictability, consistency, or even diversity
A graph may not only show the mean and the standard deviation, but also uses these two values to:
Reconstruct the underlying scale of measurement (the X values along the horizontal line)
Sample standard deviation is:
Represented by the symbol s and equals the square root of the sample variance
Sample variance is:
Represented by the symbol s^2 and equals the mean squared distance from the mean. Sample variance is obtained by dividing the sum of squares (SS) by n − 1.
Population standard deviation is:
Represented by the symbol σ and equals the square root of the population variance
Population variance is:
Represented by the symbol σ^2 and equals the mean squared distance from the mean. Population variance is obtained by dividing the sum of squares (SS) by N.
The symbol __ is used for the sample standard deviation
SD
Sometimes the table also indicates the:
Sample size, n, for each group
remember that knowing the sample mean places a restriction on:
Sample variability-only n − 1 of the scores are free to vary; df = n - 1
The n − 1 degrees of freedom for a sample is the same n − 1 that is used in the formulas for:
Sample variance and standard deviation
The concepts of variance and variance standard deviation are the same for both:
Samples and populations (however, the details of the calculations differ slightly, depending on whether you have data from a sample or from a complete population)
The basic assumption of this process of inferential statistics is that:
Samples should be representative of the populations from which they come
Although the concept of the range is fairly straightforward, there are:
Several distinct methods for computing the numerical value
To identify the standard deviation, we use the Greek letter:
Sigma (lower case): σ for population standard deviation
The basic question is whether the patterns observed in the sample data reflect corresponding patterns that exist in the population, or are:
Simply random fluctuations that occur by chance
If there are small differences between scores, then the variability is _____, and if there are large differences between scores, then the variability is _____
Small, large
Notice that the relative position of a score depends in part on the size of the:
Standard deviation
The _____ is the most commonly used and the most important measure of variability
Standard deviation
Of the 3 measures of variability, which 2 are most important?
Standard deviation and the related measure of variance
When reporting the descriptive measures for several groups, the findings may be summarized in a:
Table
Conceptually, however, the standard deviation provides a measure of:
The average distance from the mean
A sample statistic is biased if:
The average value of the statistic either underestimates or overestimates the corresponding population parameter
A sample statistic is unbiased if:
The average value of the statistic is equal to the population parameter. (The average value of the statistic is obtained from all the possible samples for a specific sample size, n.)
What are the changes in notation for a sample?
The changes in notation involve using M for the sample mean instead of μ, and using n (instead of N) for the number of scores
Thus, predictability, consistency, and diversity are all concerned with:
The differences between scores or between individuals, which is exactly what is measured by variability
What happens to the standard deviation when the scores are transformed in this manner?
The easiest way to determine the effect of a transformation is to remember that the standard deviation is a measure of distance. If you select any two scores and see what happens to the distance between them, you also will find out what happens to the standard deviation.
What is the effect of this adjustment?
The effect of the adjustment is to increase the value you will obtain. Dividing by a smaller number (n − 1 instead of n) produces a larger result and makes sample variance an accurate and unbiased estimator of population variance
In general, with a sample of n scores, the first n − 1 scores are free to vary, but:
The final score is restricted (ex. n=3, n-1=3-1=2, therefore 2 of 3 scores are free to very, but last one is restricted)
Since the standard deviation equations are complex, we begin by looking at:
The logic that leads to these equations
Rather than listing all of the individual scores in a distribution, research reports typically summarize the data by reporting only:
The mean and the standard deviation (when you are given these two descriptive statistics, however, you should be able to visualize the entire set of data)
A frequency distribution histogram for a population of N=5 scores:
The mean for this population is μ=6. The smallest distance from the mean is 1 point and the largest distance is 5 points. The standard distance (or standard deviation) should be between 1 and 5 points.
When the scores are whole numbers, the range can also be defined as:
The number of measurement categories
The degrees of freedom determine:
The number of scores in the sample that are independent and free to vary
The bias of sample variability:
The population of adult heights forms a normal distribution. If you select a sample from this population, you are most likely to obtain individuals who are near average in height. As a result, the variability for the scores in the sample is smaller than the variability for the scores in the population.
What is a problem with using the range as a measure of variability?
The problem with using the range as a measure of variability is that it is completely determined by the two extreme values and ignores the other scores in the distribution-thus, a distribution with one unusually large (or small) score will have a large range even if the other scores are all clustered close together
The obvious first step toward defining and measuring variability is:
The range
Because the value of the sample mean varies from one sample to another, you must first compute:
The sample mean before you can begin to compute deviations
The frequency distribution histogram for a sample of n=8 scores:
The sample mean is M=6.5. The smallest distance from the mean is 0.5 points, and the largest distance from the mean is 4.5 points. The standard distance (standard deviation) should be between 0.5 and 4.5 points, or about 2.5.
An alternative definition of the range is often used when:
The scores are measurements of a continuous variable (in this case, the range can be defined as the difference between the upper real limit (URL) for the largest score and the lower real limit (LRL) for the smallest score)
Technically, the standard deviation is:
The square root of the average squared deviation
In simple terms, the standard deviation provides a measure of:
The standard, or average, distance from the mean, and describes whether the scores are clustered closely around the mean or are widely scattered
Standard deviation provides a measure of:
The typical, or standard, distance from the mean
Specifically, with a population, you find the deviation for each score by measuring its distance from the population mean. With a sample, on the other hand:
The value of μ is unknown and you must measure distances from the sample mean
Variability plays an important role in the inferential process because:
The variability in the data influences how easy it is to see patterns
In simple terms, if the scores in a distribution are all the same, then:
There is no variability
Because the mean is a critical component in the calculation of standard deviation and variance, the same restrictions that apply to the mean also apply to:
These two measures of variability
To say that things are variable means that:
They are not all the same
The definitional formula provides the most direct representation of the concept of SS; however:
This formula can be awkward to use, especially if the mean includes a fraction or decimal value
When does a transformation of a set of scores occur?
This happens, for example, when exposure to a treatment adds a fixed amount to each participant's score or when you want to change the unit of measurement (to convert from minutes to seconds, multiply each score by 60)
What is error variance?
This term is used to indicate that the sample variance represents unexplained and uncontrolled differences between scores
How do we correct for this problem or sample variability underestimating population variability?
To correct for this problem we adjusted the formula for sample variance by dividing by n − 1 instead of dividing by n
In summary, both the sample mean and the sample variance (using n − 1) are examples of:
Unbiased statistics
Earlier we noted that sample variability tends to _____ the variability in the corresponding population
Underestimate
This bias is in the direction of:
Underestimating the population value rather than being right on the mark.
How can we fix this issue of the definitional formula being awkward?
Use the computational formula
The goal of inferential statistics is to:
Use the limited information from samples to draw general conclusions about populations
This assumption of inferential statistics poses a special problem for:
Variability because samples consistently tend to be less variable than their populations
Although the concept of standard deviation is straightforward, the actual equations tend to be more complex and lead us to the related concept of:
Variance before we finally reach the standard deviation
In the same way that sum of squares, or SS, is used to refer to the sum of squared deviations, the term mean square, or MS, is often used to refer to:
Variance, which is the mean squared deviation
For the population, a deviation of 4 points from the mean is relatively small, corresponding to only 1/2 of the standard deviation. For the sample, on the other hand, a 4-point deviation is:
Very large, equaling twice the size of the standard deviation
In addition to describing distributions of scores, variability also helps us determine:
Which outcomes are likely and which are very unlikely to be obtained (plays an important role in inferential statistics)
However, these extreme values of the population are unlikely to be obtained when:
You are selecting a sample, which means that the sample variability is relatively small
For a sample of n scores, the degrees of freedom, or df, for the sample variance are defined as:
df = n - 1
To emphasize the relationship between standard deviation and variance, we use __ as the symbol for population variance (standard deviation is the square root of the variance)
σ^2
