### One variable statistics measures of spread part 1 independent

The story before this unit: In grades 6 to 8, students were introduced to data sets and different ways to represent data histograms, dot plots, box plots. Statistics is introduced as a tool to answer questions about a population that have variability in the answer. Students learn about measures of center median, mean and measures of variability interquartile range, mean absolute deviationusing them to draw informal comparative inferences about two populations. The part of the story happening in this unit: Students build on and expand their understandings of statistics in this unit.

The key characteristics measures of shape, center, and spread are again seen and in addition, students may further describe the shape of a data distribution symmetric, skewed, flat or bell shaped and summarize by a statistic measuring center and a statistic measuring spread. Instead of creating representations of data, the emphasis in high school is on interpreting representations and judiciously interpreting measures of center and spread. Students develop more precise understanding of measures of center.

They learn that mean and median are equal for symmetrical distributions, explain why mean and median are not equal in examples of skewed distributions, select median as the better measure of center for skewed distributions, and make generalizations about what kinds of distributions have means larger than medians and which have medians larger than means.

To aid in developing their understanding, students will calculate a standard deviation by hand for a small data set at least once. Given different visual representations of data box plots, histograms, dot plots students draw and justify significant and meaningful conclusions about the given situation.

In unit S2 which could take place either before or after this unitstudents also build their statistics foundation by learning ways to determine whether two sets of data are correlated, and how strongly. Students identify linear association and interpret slope and intercept in the context of the data. Students are introduced to two-way frequency tables and understand how to interpret relative frequencies in the context of the data represented in the tables.

The story after this unit: In unit A1. Given different visual representations of data linear models students draw and justify significant and meaningful conclusions about the given situation. Students begin to use technology as a means to plot data and generate correlation coefficients.

In unit A2. They learn about normal distributions and use them to solve problems, and use the distributions of probability models to find the likelihood of a particular outcome. In doing so, students build on their experience with standard deviations from S1, calculating standard deviations using technology, and interpreting the results.

Every high school statistics and probability standard is a modeling standard, hence modeling pervades the four units. Is there something weird about it that means we should disregard it S-ID. Course High School - A1.

Unit One Variable Statistics. Unit: A1. Sections A1. View Details. View Full Details.Cases are also sometimes known as units or experimental units. In other words, something that can vary. Data are collected from a sample of STAT students. A third grade teacher wants to know if students who spend more time studying at home get higher homework and exam grades. A researcher wants to know if dogs who are fed only canned food have different body mass indexes BMI than dogs who are fed only hard food.

They collect BMI data from 50 dogs who eat only canned food and 50 dogs who eat only hard food. Breadcrumb Home 1 1. Case An experimental unit from which data are collected.

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### One Variable Statistics

Help F1 or? Save changes Close.So far we have learned about different ways to quantify the center of a distribution. A measure of center by itself is not enough, though, to describe a distribution.

Consider the following two distributions of exam scores. Both distributions are centered at 70 the median of both distributions is approximately 70but the distributions are quite different. In order to describe the distribution, we therefore need to supplement the graphical display not only with a measure of center, but also with a measure of the variability or spread of the distribution.

Although the measures of center did approach the question differently, they do attempt to measure the same point in the distribution and thus are comparable. However, the three measures of spread provide very different ways to quantify the variability of the distribution and do not try to estimate the same quantity. In fact, the three measures of spread provide information about three different aspects of the spread of the distribution which, together, give a more complete picture of the spread of the distribution.

The range is exactly the distance between the smallest data point min and the largest one Max. The following picture illustrates this idea: Think about the horizontal line as the data ranging from the min to the Max. The equal distances indicate equal amounts of data NOT equal distance between the numeric values. Although we will use software to calculate the quartiles and IQR, we will illustrate the basic process to help you fully understand. Q1 is the median of the bottom half of the data.

Since there are 16 observations in that half, Q1 is the mean of the 8th and 9th ranked observations in that half:. Similarly, Q3 is the median of the top half of the data, and since there are 16 observations in that half, Q3 is the mean of the 8th and 9th ranked observations in that half:.

Looking again at the histogram will illustrate this:. Q1 and Q3 as reported by the various software packages differ from each other and are also slightly different from the ones we found here. This should not worry you. There are different acceptable ways to find the median and the quartiles. We also noted that the IQR should be paired as a measure of spread with the median as a measure of center.

The idea behind the standard deviation is to quantify the spread of a distribution by measuring how far the observations are from their mean. The standard deviation gives the average or typical distance between a data point and the mean. In order to get a better understanding of the standard deviation, it would be useful to see an example of how it is calculated.Unlike linear regression, multiple regression simultaneously considers the influence of multiple explanatory variables on a response variable Y.

In other words, it permits us to evaluate the effect of more than one independent variable on a given dependent variable. Normally, statistical software such as Excel and R are used to estimate the multiple regression model. The interpretation of the multiple regression coefficients is quite different compared to linear regression with one independent variable.

The effect of one variable is explored while keeping other independent variables constant. The slope coefficient, in this case, is 0. A unit increase in X 1 will not result in a 0. Therefore, we will interpret 0. Although the multiples regression parameters can be estimated, it is challenging since it involves a huge amount of algebra and the use of matrices. We can, however, build a foundation of understanding using the multiple regression model with two explanatory variables.

The next step is to regress Y on X 2 to get the residuals of Y i, which is intuitively given by:. The last step regression gives the regression between the components of Y and X 1, which is uncorrelated with X 2.

By repeating this process, we can estimate a k-parameter model such as:. Suppose that we have n observations of the dependent variable Y and the independent variables X 1X 2. For us to make valid inference from the above equation, we need to make classical normal multiple linear regression model assumptions as follows:. The assumptions are almost the same as those of linear regression with one independent variable, only that the second assumption is tailored to make sure that there are no linear relationships between the independent variables multicollinearity.

Recall that the standard error estimate gives a percentage at which we are certain of a forecast made by a regression model. However, it does not tell us how suitable is the independent variable in determining the dependent variable.

The coefficient of variation corrects this shortcoming. The coefficient of variation measures a proportion of the total change in the dependent variable that is explained by the independent variable. We can calculate the coefficient of variation in two ways:. The coefficient of variation can be computed by squaring the correlation coefficient r between the dependent and independent variables. That is:. The correlation coefficient between the money supply growth rate dependent, Y and inflation rates independent, X is 0.

The standard deviation of the dependent explained variable is 0. Regression analysis for the ten years was conducted on this variable. We need to calculate the coefficient of determination. So, in regression, the money supply growth rate explains roughly If the regression analysis is known, then our best estimate for any observation for the dependent variable would be the mean. Alternatively, instead of using the mean as an estimate of Y iwe can predict an estimate using the regression equation.

The resulting solution will be denoted as:. The expression, therefore, reduces to. If the regression analysis is useful for predicting Y i using the regression equation, then the error should be smaller than predicting Y i using the mean. Now, recall than the coefficient of determination is the fraction of the overall change that is reflected in the regression. Denoted by R 2coefficient of determination is given by:.

Other values are in the range of 0 and 1 and always positive. The adjusted R-squared can increase, but that happens only if the new variable improves the model more than would be expected by chance. If the added variable improves the model by less than expected by chance, then the adjusted R-squared decreases. Previously, we had conducted hypothesis tests on individual regression coefficients using the t-test.Please remember to click the Submit button for each separate question, and read the feedback comments!

Which of the following are measures of dispersion? The correct answers are bde and i. A measure of dispersion quantifies the spread of the variable. A measure of dispersion is often presented alongside an appropriate measure of central tendency. The heights cm of 6 children were measured as, The correct answer is d.

The range is the difference between the minimum and the maximum. The minimum is and the maximum is What is the standard deviation of the data? Use Use Excel or a calculator. The correct answer is a. The standard deviation shows how much variation or dispersion there is from the mean. It is expressed in the same units at the data. The standard deviation in this case is 8. Click here to see the formula for a sample standard deviation s.

As in the previous questions, the heights cm of 6 children were measured as,What is the variance of the data? The correct answer is c. The variance is the square of the standard deviation. Therefore the variance is The correct answer is h.

The bar chart is used to display the frequency of a categorical variable. This categorical variable is purely nominal. It does not make sense to report a measure of dispersion for a variable of this type. The correct answers are acd and e.Liga Belgium First Division A First Division B Bulgaria Croatia Cyprus Czech Republic Denmark Superliga 1st Division England Premier League Championship League One League Two National League Estonia Finland France Ligue 1 Ligue 2 National Germany Bundesliga 2.

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**Choosing which statistical test to use - statistics help.**

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## Measures of Spread

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