Thirty-six lasted three days. Eighteen lasted four days. Nineteen lasted five days. Four lasted six days. One lasted seven days. One lasted eight days. One lasted nine days. A survey of enrollment at 35 community colleges across the United States yielded the following figures:. Use the following information to answer the next two exercises. The number that is 1. Suppose that a publisher conducted a survey asking adult consumers the number of fiction paperback books they had purchased in the previous month.
The results are summarized in the Table. The standard deviation provides a numerical measure of the overall amount of variation in a data set, and can be used to determine whether a particular data value is close to or far from the mean. The standard deviation provides a measure of the overall variation in a data set The standard deviation is always positive or zero. Rosa waits for seven minutes: Seven is two minutes longer than the average of five; two minutes is equal to one standard deviation.
Rosa's wait time of seven minutes is two minutes longer than the average of five minutes. Rosa's wait time of seven minutes is one standard deviation above the average of five minutes. Binh waits for one minute. One is four minutes less than the average of five; four minutes is equal to two standard deviations.
Binh's wait time of one minute is four minutes less than the average of five minutes. Binh's wait time of one minute is two standard deviations below the average of five minutes. A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if it is more than two standard deviations away is more of an approximate "rule of thumb" than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is further away than two standard deviations.
You will learn more about this in later chapters. Sampling Variability of a Statistic The statistic of a sampling distribution was discussed previously in chapter 2.
The ages are rounded to the nearest half year: 9; 9. Data Freq. Deviations Deviations 2 Freq. Verify the mean and standard deviation or a calculator or computer. Find the value that is one standard deviation above the mean. Find the value that is two standard deviations below the mean. Find the values that are 1. This example can help us get ready for finding standard deviations of frequency distributions, so we'll emulate what was done above in the spreadsheet.
Using the table above instead of the raw data, put the data values 9, 9. We can take advantage of cell references to avoid typing repeated numbers and possibly making mistakes. We'll essentially copy the table above in the spreadsheet, but select the cells instead of typing them in. We can make the Spreadsheet do the calculations for us. For a number we don't want to change the mean in this case , we can "lock" the cell reference using dollar signs around the letter.
In this example, the mean is located in cell A9. Enter 2nd 1 for L1, the comma , , and 2nd 2 for L2. Enter data into the list editor. If necessary, clear the lists by arrowing up into the name. Put the data values 9, 9. Use the arrow keys to move around. Press VarStats and enter L1 2nd 1 , L2 2nd 2.
Do not forget the comma. Exercise 2. Explanation of the standard deviation calculation shown in the table The deviations show how spread out the data are about the mean. Make comments about the box plot, the histogram, and the chart. Endpoints of the intervals are as follows: the starting point is Standard deviation of Grouped Frequency Tables Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision.
Spreadsheets For the previous example, we can use the spreadsheet to calculate the values in the table above, then plug the appropriate sums into the formula for sample standard deviation. Comparing Values from Different Data Sets The standard deviation is useful when comparing data values that come from different data sets. For each data value, calculate how many standard deviations away from its mean the value is. References Data from Microsoft Bookshelf.
King, Bill. Available online at www. This type of error is considered a more serious problem because it may result in and under- or overestimation of the true association. Because of the play of chance, different samples will produce different results and therefore this must be taken into account when using a sample to make inferences about a population.
Sampling error cannot be eliminated but with an appropriate study design it can be reduced to an acceptable level. One of the major determinants of the degree to which chance can affect the findings of a study is the sample size.
Therefore, use of an appropriate sample size will reduce the degree to which chance variability may account for the results observed in a study. The role of chance can be assessed by performing appropriate statistical tests to produce a p-value and by calculation of confidence intervals. Confidence intervals are more informative than p-values because they provide a range of values that is likely to include the true population effect.
They also indicate whether a non-significant result is, or is not, compatible with a true effect that was not detected because the sample size was too small. NB: Statistical methods only assess the effect of sampling variation and cannot control for non-sampling errors such as confounding or bias in the design, conduct or analysis of a study. Sources of variation in measurements 3. The quality of measurement data is vital for the accurate classification of study participants according to their personal attributes, exposure and outcome.
Unlike studies involving routine data, which has already been collected, investigators carrying out their own measurements have the advantage of being able to choose which observations they will make, and to maximise the quality of their data.
However, each measurement will usually only be made once and it is vital that every effort is made to ensure consistent results are obtained between patients. Differences made on the same subject on different occasions may be due to several factors, including:. Variations in recording observations arise for several reasons including bias, errors, and lack of skill or training. There are two principal types:.
Avoiding variation in measurements. Prior to starting data collection, careful thought should be given to potential sources of error, bias and variation in measurements, and every effort made to minimise them.
The coefficient of variation CV is a statistical measure of the dispersion of data points in a data series around the mean. The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.
The coefficient of variation shows the extent of variability of data in a sample in relation to the mean of the population. In finance, the coefficient of variation allows investors to determine how much volatility, or risk, is assumed in comparison to the amount of return expected from investments. Ideally, if the coefficient of variation formula should result in a lower ratio of the standard deviation to mean return, then the better the risk-return trade-off.
Note that if the expected return in the denominator is negative or zero, the coefficient of variation could be misleading. For example, an investor who is risk-averse may want to consider assets with a historically low degree of volatility relative to the return, in relation to the overall market or its industry. Conversely, risk-seeking investors may look to invest in assets with a historically high degree of volatility.
While most often used to analyze dispersion around the mean, quartile, quintile, or decile CVs can also be used to understand variation around the median or 10th percentile, for example. The coefficient of variation formula or calculation can be used to determine the deviation between the historical mean price and the current price performance of a stock, commodity, or bond, relative to other assets.
Below is the formula for how to calculate the coefficient of variation:. Please note that if the expected return in the denominator of the coefficient of variation formula is negative or zero, the result could be misleading.
The coefficient of variation formula can be performed in Excel by first using the standard deviation function for a data set.
Next, calculate the mean using the Excel function provided. Since the coefficient of variation is the standard deviation divided by the mean, divide the cell containing the standard deviation by the cell containing the mean. Doing so provides needed context, points to opportunity, and helps them maintain their cool when something goes wrong.
Consider the following example. The figure below depicts the error rates for the first three weeks of an invoicing process:. After week two, the responsible manager was embarrassed — could her team really be performing that poorly? After the third, she breathed a sigh of relief.
The error rate may be high, but at least the trend was in the right direction! Unfortunately, her interpretation did not hold up. Here are the measurements for the next seven weeks:. Her mistake arose because she did not understand that all processes vary, often considerably! This vignette underscores the first point, which is simply to acknowledge that variation is important and take it into account.
For instance, everyone knows that some full-grown adults are taller than others, and it is easy enough to observe that men, on average, are taller than women.
So, in this instance, one component of variation is gender. Similarly, people from the Netherlands are generally taller, and those from the Philippines are generally shorter.
Nationality, then, is another source of variation. These sources become increasingly important as you gain a feel for measurements of variation.
Instead, focus on interpretation.
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