Statistics for People Who (Think They) Hate Statistics

Boldface represents concept name, underline means materials, asterisk* represents advanced concept, and

1. number listed text is central to the topic.

###20201123 Chapter 1

Infants' Growth: Lampl, M., Veldhuis, J. D., & Johnson, M. L. (1992). Saltation and stasis: A model of human growth. Science, 258, 801-803.

Educational television and children's psychological development: Collins, P. A., Wright, J. C., Anderson, R., Huston, A. C., Schmitt, K., & McElroy, E. (1997, April). Effects of early childhood media use on adolescent achievement.

Statistics - a set of tools and techniques that are used for describing, organizing, and interpreting information or data..

Descriptive Statistics - are used to organize and describe the characteristics of a collection of data. The collection is sometimes called a data set or just data. For example, the mode, the mean, etc.

Inferential Statistics - are used to make inferences based on a smaller group of data (such as our group of 22 students) about a possibly larger one (such as all the undergraduate students in the College of Arts and Sciences). They are often (but not always) the next step after you have collected and summarized data.

A smaller group of data is often called a sample, which is a portion, or a subset, of a population.

###20201124/25/26 Chapter 2

Study Site: edge.sagepub.com/salkind6e

An average - is the one value that best represents an entire group of scores. Averages, also called measures of central tendency, include the mean, the median, and the mode. Each provides you with a different type of information about a distribution of scores.

The mean - is simply the sum of all the values in a group, divided by the number of values in that group. It is the most common type of average that is computed.

1. The mean is sometimes represented by the letter M and is also called the typical, average, or most central score.
2. A small n represents the sample size for which the mean is being computed. A large N would represent the population size. In some readings, no distinction is made between the two.
3. The sample mean is the measure of central tendency that most accurately reflects the population mean.
4. Finally, the mean is very sensitive to extreme scores. An extreme score can pull the mean in one or the other direction and make it less representation of the set of scores and less useful as a measure of central tendency.

*arithmetic mean/harmonic mean

A weighted mean - can be easily computed by multiplying the value by the frequency of its occurrence, adding the total of all the products, and then dividing by the total number of occurrences.

In basic statistics, an important distinction is made between those values associated with samples (a part of a populations) and those associated with populations. For a sample statistic (such as the mean of a sample), Roman letters are used. For a population parameter (such as the mean of a population), Greek letters are used.

The median - is defined as the midpoint in a set of scores.

*Percentile points - are used to define the percentage of cases equal to and below a certain point in a distribution or set of scores. If a score is "at the 75th percentile," it means that the score is at or above 75% of the other scores in the distribution. The median is also known as the 50th percentile. Q1 25th percentile, Q3 75th percentile.

Why use the median instead of the mean? i.e. certain 某些 social and economic indicators, median family income Compared to the mean, the median is insensitive to extreme scores. When you have a set of scores in which one or more scores are extreme, the median better represents the centermost value of that set of scores than any other measure of central tendency. There are just too many extreme scores that would skew, or significantly distort.

Skew - is the quality of a distribution that defines the disproportionate frequency of certain scores. A longer right tail than left corresponds to a smaller number of occurrences at the high end of the distribution; this is a positively skewed distribution. A shorter right tail than left corresponds to a larger number of occurrences at the high end of the distribution; this is a negatively skewed distribution.

Just like M for the mean, Med or Mdn for median.

1. While the mean is the middle point of a set of values, the median is the middle point of a set of cases.
2. Because the median cares about how many cases there are, not the values of those cases, extreme scores (sometimes called outliers) don't count 有价值.

The mode - is the value that occurs most frequently. It is the most general and least precise measure of central tendency, but it plays a very important part in understanding the characteristics of a set of scores.

Selecting the label of the category itself, rather than the number of times a category occurs.

If every value in a distribution contains the same number of occurrences, then there really isn't a single mode. But if more than one value appears with equal frequency, the distribution is multimodal. The set of scores can be biomodal (with two modes).

Measurement - is the assignment of values to outcomes following a set of rules - simple.按照一组规则（简单）将值分配给结果。

These scales of measurement, or rules, are the particular levels at which outcomes are observed. Each level has a particular set of characteristics, and scales of measurement come in four flavors: nominal, ordinal, interval, and ratio.

Source: Coursera, Basic Statistics by University of Amsterdam

A rose by any other name The nominal level of measurement - is defined by the characteristics of an outcome that fit into one and only one class or category. Nominal-level variables are "names" (nominal in Latin), and the nominal level can be the least precise level of measurement. i.e. gender (female and male), ethnicity (Caucasian or African American), political affiliation (Republican, Democrat, or Independent).

Any order is fine with me The ord in ordinal level of measurement - stands for order, and the characteristic of things being measured here is that they are ordered. i.e. a rank of candidates for a job (Russ ranks #1, Sheldon ranks #2, Hannah ranks #3), no idea how much higher on this scale Russ is relative to Sheldon than Sheldon is relative to Hannah.

1+1=2 The interval level of measurement - is a test or an assessment tool is based on some underlying continuum such that we can talk about how much more a higher performance is than a lesser one. The intervals or spaces or points along the scale are equal to one another, 10 words correct is 2 more than 8 correct, which is 3 more than 5 correct. i.e. If you get 10 words correct on a vocabulary test, that is twice as many as getting 5 words correct.

Can anyone have nothing of anything The ratio level of measurement - is characterized by the presence of an absolute zero on the scale. What that zero means is the absence of any of the trait that is being measured. *The conundrum - are there outcomes we measure where it is possible to have nothing of what is being measured? In the physical and biological sciences, you can have the absence of a characteristic, such as absolute zero (no molecular movement) or zero light. In the social and behavioral sciences, it's a bit harder. Even if you score zero on that spelling test or miss every item of an IQ test (in Russian), that does not mean that you have no spelling ability or no intelligence, right?

1. Any outcome can be assigned to one of four scales of measurement.
2. Scales of measurement have an order, from the least precise being nominal to the most precise being ratio.
3. The "higher up" the scale of measurement, the more precise the data being collected, and the more detailed and informative the data are.
4. Finally, the more precise scales contain all the qualities of the scales below them.

Mode is used for categorical data in nature, that means values can fit into only one class, such as hair color, sex, etc. These categories are called mutually exclusive.

Median is used when there are extreme scores.

Mean is the most often used measure of central tendency. It is a more precise measure than the median and that the median is a more precise measure than the mode.

###20201126/27 Chapter 3

Variability (also called spread or dispersion)- reflects how scores differ from one another. Three measures of variability are commonly used to reflect the degree of variability, spread, or dispersion in a group of scores. These are the range, the standard deviation, and the variance.

The range - means how far apart scores are from one another. It is the most general measure of variability.

*exclusive range (h-l)/inclusive range (h-l+1)

The standard deviation (abbreviated as s or SD) - represents the average amount of variability in a set of scores. In practical terms, it's the average distance from the mean. The larger the standard deviation, the larger the average distance each data point is from the mean of the distribution and the more variable the set of scores is.

*The mean deviation (also called the mean absolute deviation) - is the sum of the absolute value of the deviations from the mean divided by the number of scores.

Why square the deviations? Because we want to get rid of the negative sign.

Why do we divide by n-1 rather than just plain ol' n? The answer is that s is an estimate of the population standard deviation, and it is an unbiased estimate at that, but only when we subtract 1 from n. By subtracting 1 from the denominator, we artificially force the standard deviation to be larger than it would be otherwise. Why would we want to do that? Because, as good scientists, we are conservative. Being conservative means that if we have to err, we will do so on the side of overestimating what the standard deviation of the population is. All other things being equal, then, the larger the size of the sample, the less difference there is between the biased and the unbiased estimates of the standard deviation. When you compute the standard deviation for a sample, which is an estimate of the population, the closer to the size of the population the sample is, the more accurate the estimate will be.

*Biased estimates are appropriate if your intent is only to describe the characteristics of the sample. But if you intend to use the sample as an estimate of a population parameter, then it's best to calculate the unbiased statistic.

1. The larger the standard deviation, the more spread out the values are, and the more different they are from one another.
2. The standard deviation is sensitive to extreme scores. When you are computing the standard deviation of a sample and you have extreme scores, note that fact somewhere in your written report and in your interpretation of what the data mean.
3. If s =0, there is absolutely no variability in the set of scores, and the scores are essentially identical in value. This will rarely happen.

The variance - is simply the standard deviation squared.

You are not likely to see the variance mentioned by itself in a journal article or see it used as a descriptive statistic. This is because the variance is a difficult number to interpret and apply to a set of data.

But the variance is important because it is used both as a concept and as a practical measure of variability in many statistical formulas and techniques.

###20201127/30 Chapter 4

Print out graph paper: www.printfreegraphpaper.com

Maintain the scale in a graph. The scale refers to the relationship between the horizontral and vertical axes. This ratio should be about 3 to 4, so a graph that is 3 inches wide will be about 4 inches tall.

A chart alone should convey what you want to say.

A frequency distribution - is a method of tallying and representing how often certain scores occur.

Here are six general rules to create a frequency distribution:

1. Determine the range.
2. Decide on the number of class intervals.
3. Decide on the size of the class interval.
4. Decide the starting point for the first class.
5. Create the class intervals.
6. Put the data into the class intervals.

A class interval - is a range of numbers.

Here are some general rules to follow in the creation of a class interval:

1. Select a class interval that has a range of 2, 5, 10, 15, or 20 data points.
2. Select a class interval so that 10 to 20 such intervals cover the entire range of data. A convenient way to do this is to compute the range and then divide by a number that represents the number of intervals you want to use (between 10 and 20).
3. Begin listing the class interval with a multiple of that interval.
4. Finally, the largest interval goes at the top of th efrequency distribution.

Histogram - is a visual representation of the frequency distribution where the frequencies are represented by bars.

A frequency polygon - is a continuous line that represents the frequencies of scores within a class interval. Place a midpoint at the top of each bar in a histogram and connect the lines.

A cummulative frequency distribution - is a visual representation of the cumulative frequency of occurrences by class intervals.

*A cumulative frequency polygon is also called an ogive. If the distribution of the data is normal or bell shaped, the ogive represents what is popularly known as a bell curve or a normal distribution. In SPSS, it's called a P-P plot (for probablity plot).

A bar chart should be used to compare the frequencies of different categories with one another. Categories are organized horizontally on the x-axis, and values are shown vertically on the y-axis.

A column chart is identical to a bar chart. But in this chart, categories are organized on the y-axis, and values are shown on the x-axis.

A line chart should be used to show a trend in the data at equal intervals.

A pie chart is used to show the proportion of an item that makes up a series of data points. Note that a pie chart only offers counts by the nominal classes (such as ethnicity, ties of enrollment and gender) the chart represents.

Kurtosis - is associated with how flat or peaked a distribution appears.

###20201201-17 Chapter 5

A correlation coefficient - is numerical index that reflects the relationship between two variables. A correlation between two variables is sometimes referred to as a bivariate (for two variables) correlation. It is called Pearson product-moment correlation.

The Pearson correlation coefficient examines the relationship between two continuous variables. Continuous variables are variables that can assume any value along some underlying continuum. i.e. height, age, test score, and income. The point-biserial correlation, or other correlational techniques, are used for discrete 离散的 or categorical variables. i.e. race, social class, political affiliation.

If variables change in the same direction, the correlation is called a direct correlation or a positive correlation. if variables change in opposite directions, the correlation is called an indirect correlation or a negative correlation.

1. The value of this descriptive statistic ranges between -1.00 and +1.00.
2. The absolute value of the coefficient reflects the strength of the correlation. So a correlation of -.70 is stronger than a correlation of +.50.
3. A correlation always reflects a situation in which there are at least two data points (or variables) per case.
4. The Pearson product-moment correlation coefficient is represented by teh small letter r with a subscript 下标 representing the variables that are being correlated. i.e. r (subscript xy)

The correlation coefficient reflects the amount of variability that is shared between two variables and what they have in common. However, if one variable does not change in value and therefore has nothing to chare, then the correlation between it and another variable is zero. Likewise, if you constrain or restrict the range of one variable, the correlation between that variable and another variable will be less than if the range is not constrained.

*parentheses n.圆括号，插入语，插曲

Scatterplot or scattergram - is simply a plot of each set of scores on separate axes. It visually represents a correlation.

The scatterplot for a strong (but not perfect) direct relationship where r(xy) = .70. The data points align themselves along a positive slope, although not perfectly.

The scatterplot for a strong indirect, or negative relationship where r(xy) = -.82. The data points align themselves on a negative slope from the upper left-hand corner of the chart to the lower right-hand corner.

Linear correlation - is reflected by a straight line showing the X and the Y values in a relationship.

*curvilinear relationship

Correlation Matrix - can illustrate correlations for more than two variables.

How to interpret the correlation coefficient?

The easiest (but not the most informative) way to interpret is by eyeballing it.

The much more precise way to interpret it: computing the coefficient of determination 确定系数.

The coefficient of determination - is the percentage of variance in one variable that is accounted for by the variance in the other variable. If the correlation between GPA and number of hours of study time is .70, then the coefficient of determination is .49. This means that 49% of the variance in GPA can be explained by the variance in studying time. The stronger the correlation, the more variance can be explained (which only makes good sense). The more two variables share in common, the more information about performance on one score can be explained by the other score.

*coefficient of alienation (or coefficient of nondetermination) - is the amount of unexplained variance.

Correlations express the association that exists between two or more variables; they have nothing to do with causality 因果关系.

Other Correlations

Partial correlation where the relationship between two variables is explored, but the impact of a third variable is removed from the relationship between the two. Sometimes that third variable is called a mediating or a confounding variable.