Introduction and Descriptive Statistics

Normal curve

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Polygon

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Provide examples of continuous variables.

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Review the formulas for variance standard deviation and other related formulas

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The computational formula for standard deviation and the sum of squares

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The definition of formula for standard deviation and the sum of squares

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The order of mathematical operations

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Verbally explain the difference between apparent limits and real limits or boundaries.

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Sample mean

Is represented by X-bar or M.

Normal distribution

Is symmetrical with the greatest frequency in the middle and smaller frequencies as you moved toward either extreme.

Standard deviation

Is the square root of the variance and provides a measure of the standard, or average, distance from the mean.

SS, or sum of squares

Is the sum of the squared deviation scores

Sigma

Is used to stand for summation. The summation sign, Sigma, can be read as "the sum of".

Symmetrical distribution

It is possible to draw a vertical line through the middle so that one side of the distribution is a mirror image of the other.

Zero variability

Scores all the same

Characteristics of the mean

1. Changing the value of any score changes the mean. 2. Adding a new scored the distribution, or removing an existing score, usually changes the mean. The exception is when the new score or their moves for is exactly equal to the mean. 3. If a constant value is added to every score in a distribution, the same constant is added to the mean. Similarly, if you subtract a constant from every school, the same constant is subtracted from the mean. 4. If every score in a distribution is multiplied by or divided by a constant value, the mean changes in the same way.

When should I use the median?

1. Extreme scores or skewed distributions. 2. Undetermined values. 3. Open-ended distributions. 4. Ordinal scale.

What does a researcher hope for in order to conclude that his hypothesis is probable?

1. Large mean difference 2. Low variability

When should I use the mode?

1. Nominal scales. 2. Discrete variables. 3. Describing shape.

What two elements does a frequency distribution always present?

1. The set of categories that make up the original measurement scale. 2. A record of the frequency, or number of individuals in each category.

Variable

A characteristic or condition that changes or has different values for different individuals.

Sample

A set of individuals selected from a population, usually intended to represent the population in a research study.

Define statistics

A set of mathematical procedures for organizing, summarizing, and interpreting data.

Datum

A single measurement or observation. Is commonly called a score or raw score

Central tendency

A statistical measure that attempts to determine the single value, usually located in the center of the distribution, that is most typical or most representative of the entire set of scores.

Parameter

A value, usually a numerical value, that describes a population. It is usually derived from measurements of the individuals in the population.

Statistic

A value, usually a numerical value, that describes a sample. It is usually derived from the individuals in the sample.

Mean

Also known as the arithmetic average, is computed by adding all the scores in the distribution and dividing by the number of scores.

Relative frequencies

Although you usually cannot find the absolute frequency for each score in a population, you very often can obtain relative frequencies. For example, you may not know exactly how many fish are in the lake, but after years of fishing you know that there are twice as many bluegill as there are bass.

Ratio scale

An interval scale with the additional feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude.

Frequency distribution

An organized tabulation of the number of individuals located in each category on the scale of measurement.

Bar graphs

Are essentially the same as a histogram, except that spaces are left between adjacent bars.

What are inferential statistics?

Consist of techniques that allow us to study samples and then make generalizations about the population from which they were selected.

Small variability

Consistent scores

Ordinal scale

Consists of a set of categories that are organized in an ordered sequence. Measurements on this scsle rank observations in terms of size or magnitude.

Nominal scale

Consists of a set of categories that have different names. Measurements on this type of scale label and categorize observations, but do not make any quantitative distinctions between observations.

Interval scale

Consists of ordered categories that are all intervals of exactly the same size. Equal differences between numbers on the scale reflect equal differences in magnitude. However, the zero point on an interval scale is arbitrary and does not indicate a zero amount of the variable being measured.

Discrete variable

Consists of separate, indivisible categories. No values can exist between two neighboring categories.

How are correlational studies different from experimental studies?

Correlational studies only require collection of information, not manipulation and control of information. Also, correlational studies cannot prove a cause-and-effect relationship between two variables, whereas experimental studies can prove a cause-and-effect relationship.

Deviation

Distance from the mean

Population variance

Equals the mean squared deviation. Variance is the average squared distance from the mean.

Population mean

Identified by the Greek letter mu

The median

If the scores in a distribution are listed from smallest to largest, the median is the midpoint of the list. More specifically, the median is the point on the measurement scale below which 50% of the scores in the distribution are located.

Quasi-independent variable

In a nonexperimental study, the "independent" variable that is used to create the different group of scores is often called the ___________.

The Correlational Method

In this method of research, two different variables are observed to determine whether there is a relationship between them.

Large variability

Inconsistent scores

Smooth curves

Indicate that you are not connecting a series of dots (real frequencies) but instead are showing the relative changes that occur from one school to the next.

Control group

Individuals who do not receive the experimental treatment. Instead, they either receive no treatment or they receive a neutral, placebo treatment. They provide a baseline for comparison with the experimental condition.

Experimental group

Individuals who do receive the experimental treatment.

What two elements are necessary for a research study to be an experiment?

Manipulation of an independent variable and rigorous control of the other extraneous variables.

Data

Measurements or observations

Proportions/relative frequencies

Measures the fraction of the total group that is associated with each score. Proportion = p = f/N

Central tendency

Measures where the center of the distribution is located.

What measure of central tendency is the most appropriate if the data is ordinal?

Median

Shape

Nearly all distributions can be classified as being either symmetrical or skewed.

Variance

Needs to be calculated in order for us to determine the standard deviation, but by itself is not very useful.

Both the standard deviation and variance are averages. Averages cannot be____________.

Negative. So, neither the standard deviation or the variance can be negative numbers.

Is it necessary to arrange the categories in a frequency distribution from highest to lowest?

No, it is customary but this is arbitrary.

The weighted mean

Often it is necessary to combine two sets of scores and then find the overall mean for the combined group. Practice the formula.

The Experimental Method

One variable is manipulated while another variable is observed and measured. To establish a cause-and-effect relationship between the two variables, an experiment attempts to control all other variables to prevent them from influencing the results.

A researcher is interested in the texting habits of high school students in the United States. If the researcher measures the number of text messages that each individual sends each day and calculates the average number for the entire group of high school students, the average number would be an example of a :

Parameter

What is one of the most common procedures for organizing a set of data?

Placing the scores in a frequency distribution

Computing the mean from a frequency distribution table

Practice.

Variability

Provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together.

Standard deviation

Represents the average difference between the scores and the mean of the distribution.

What are the three characteristics that completely describe any distribution?

Shape, central tendency, and variability.

What are descriptive statistics?

Statistical procedures used to summarize, organize, and simplify data.

Variability

Tells whether the scores are spread over a wide range or are clustered together.

Real limits

The boundaries of intervals for scores that are represented on a continuous number line. The real limit separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits. The upper real limit is at the top of the interval, and the lower real limit is at the bottom.

Range

The distance covered by the scores in a distribution, from the smallest score to the largest score.

Population

The entire set of the individuals of interest for a particular research question

What measure of central tendency is adversely affected by extreme outlier scores?

The mean

What measure of central tendency should we use if the distribution is normal?

The mean or the average

What measure of central tendency is the best if the distribution is skewed?

The median

What measure of central tendency should we use if the distribution is open-ended?

The median

Sampling error

The naturally occurring discrepancy, or error, that exists between a sample Statistic and the corresponding population parameter.

Raw score

The original, unchanged scores obtained in the study. Scores for a particular variable are represented by the letter X. When two variables are measured for each individual, the data can be presented as two lists labeled X & Y.

What are some limitations of the range?

The range is adversely affected by extreme scores.

The mode

The score or category in a frequency distribution that has the greatest frequency.

Skewed distribution

The scores tend to pile up toward one end of the scale and taper off gradually at the other end.

Tail

The section where the scores taper off is called the blank of the distribution.

Negatively skewed

The tail is on the left hand side and point toward the negative side of the x axis.

Positively skewed

The tail is on the right hand side and points to the positive end of the x-axis.

Independent variable

The variable that is manipulated by the researcher.

The Dependent variable

The variable that is observed to assess the effect of the treatment.

Mode > median > meam

Then, the distribution is negatively skewed.

Mean > median > Mode

Then, the distribution is positively skewed.

Mean = median = mode

Then, the distribution is symmetrical.

Continuous variable

There are an infinite number of possible values that fall between any two observed values. It is divisible into an infinite number of fractional parts.

Histograms

To construct a histogram, you first list the numerical scores (the categories of measurement) along the x-axis. Then you draw a bar above each x value so that: A.) The height of the bar corresponds to the frequency for that category. B.) For continuous variables, the width of the bar extends to the real limits of the category. For discrete variables, the width of the bar extends exactly half the distance to the adjacent category on each side.

What are the limits of the Correlational method?

While the results of a Correlational study can demonstrate the existence of a relationship between two variables, it cannot demonstrate a cause-and-effect relationship.

Provide some examples of discrete variables.

Whole countable, numbers; such as the number of children in a family or the number of students attending class.

What measure of central tendency can you find for qualitative data?

You can find the mode, but not the median or mean.