8.5 variance summary 3

Lesson 8: Multivariate Analysis of Variance MANOVA

To test for the effects of drug type, we give coefficients with a negative sign for drug A, and positive signs for drug B. Because there are two doses within each drug type, the coefficients take values of plus or minus 1/2. Differences among treatments can be explored through pre-planned orthogonal contrasts. Contrasts involve linear combinations of group mean vectors instead of linear combinations of the variables.

The standard deviation provides a measure of the overall variation in a data set

An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.

A common experimental design that violates Assumption 3 isto use the same unit in each group. Assumption 1 is typical of parametric statistics, because it provides a way to identify the data with known distributions. With a significance level of 5%, test the claim that a single line causes lower variation among waiting times (shorter waiting times) for customers. Math instructors are not only interested in how their students do on exams, on average, but how the exam scores vary. To many instructors, the variance (or standard deviation) may be more important than the average. Here we will sum over the treatments in each of the blocks so the dot appears in the first position.

Pay careful attention to signs when comparing and interpreting the answer. If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by \(N\), the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n – 1, one less than the number of items in the sample. Because supermarket B has a higher standard deviation, we know that there is more variation in the wait times at supermarket B.

Remember, variances are expressed at the absolute values meaning we do not show negative or positive numbers. We express variances in terms of FAVORABLE or UNFAVORABLE and negative is not always bad or unfavorable and positive is not always good or favorable. In this data set, cows in Davis, CA were sprayed with water on hot days to try to cool them down.

• Based on this plot, it appears there may be a difference in mean grey score by population.
• The segment includes glacier-sourced mineral water, high-altitude spring water, and select imported premium labels positioned to meet the expectations of urban households.
• The boxplot below is based on the birth weights of infants with severe idiopathic respiratory distress syndrome (SIRDS).
• Rising health awareness, lifestyle upgrades, and a growing focus on sustainability further support the growth of China premium bottled water industry.
• The final measure of variability we must consider are the quartiles.
• Create a simulation that shows that the one-way ANOVA test statistic follows an \(F\) distribution under the null hypothesis.

Formula for Sample Variance

In a sense, they calculate the same thing but are expressed differently, and the standard deviation is usually considered easier to interpret. If you specify a particular order, you can compute this trivially in R (e.g. via the update and anova functions, see below), but a different order of entry would yield potentially very different answers. The mineral water segment accounted for a revenue share of 41.5% in 2024. This growth is driven by strong consumer preference for bottled water sourced from trusted natural springs and mountain reserves, which offer certified purity and balanced mineral content.

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See below for a summary of the six variances from standard discussed in this chapter. The supermarkets & hypermarkets accounted for the largest revenue share in 2024. This growth is driven by their extensive reach, 8.5 variance summary strong consumer trust, and ability to showcase a wide selection of premium domestic and imported brands under one roof. These retail formats offer high visibility for premium bottled water products through dedicated shelf space, branded displays, and promotional activities that attract households seeking convenience and product variety. Strategic partnerships with major supermarket chains and hypermarket operators continue to strengthen market penetration for leading brands, ensuring consistent availability and reinforcing consumer preference for trusted retail channels.

5: Describe How Companies Use Variance Analysis

Since you will use technology to find these, the distribution and the test statistic will not be presented. Note that to obtain a statistically significant result there need only be a difference between any two of the k means. Another possibility is that management may have built the favorable variance into the standards. Management may overestimate the material price, labor rate, material quantity, or labor hours per unit, for example.

Formula Review

• Although the demand for all chocolate has beenincreasing, consumer tastes have been gradually shifting towardsdark chocolate because of its purported healthbenefits.
• The built-in data set morley gives speed of light measurements for five experiments done by Michelson in 1879.
• Suppose a sample of 15 ISPs is taken, and the standard deviation is 13.2.
• If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by \(N\), the number of items in the population.

To find the test statistic and p-value using the TI-83/84, type each data set into L1 through L5. Then go into STAT and over to TESTS and choose ANOVA(. Then type in L1,L2,L3,L4,L5 and press enter. You will get the results of the ANOVA test. There appears to be a difference between at least two of the means, but realize that the standard deviations are very different. Now before conducting the hypothesis test, look at the means and standard deviations. SSP stands for the sum of squares and cross products discussed above. The last two columns look at the absolute values of the deviations rather than the deviations themselves.

A correctly functioning test would have type I error of 0.05 in this setting. Things are worse when the third group has large variance, not quite as bad when it has small variance. The equal variances assumption of ANOVA is often difficult to verify. In this section, we introduce a variant of one-way ANOVA that corrects for unequal variance. We then use simulation to explore the effect of unequal variance on the results of one-way ANOVA. We will first examine the data to see whether it appears to be approximately normal with equal variances across the groups.

China Premium Bottled Water Market Report Scope

The online segment is expected to grow at the fastest CAGR over the forecast period. This growth is supported by the rapid expansion of e-commerce platforms across major cities and the efforts of premium bottled water brands to strengthen digital distribution. Companies partner with major online marketplaces and offer direct-to-consumer services to reach urban households that prioritize convenience, flexible purchase options, and wider product choices. The sparkling water segment is expected to grow at the fastest CAGR of 9.1% from 2025 to 2033, driven by a shift in consumer preferences toward beverages that offer health benefits and a premium, refreshing experience. Urban consumers, especially younger groups, prefer sparkling mineral water as an alternative to sugary soft drinks, which reflects stronger health awareness and demand for low-calorie choices.

Management can use standard costs to prepare the budget for the upcoming period, using the past information to possibly make changes to production elements. Standard costs are a measurement tool and can thus be used to evaluate performance. As you’ve learned, management may manage “to the variances” and can manipulate results to meet expectations. To reduce this possibility, performance should be measured on multiple outcomes, not simply on standard cost variances.

Thus, we define standard deviation as the “spread of the statistical data from the mean or average position”. Managers sometimes focus only on making numbers for the current period. For example, a manager might decide to make a manufacturing division’s results look profitable in the short term at the expense of reaching the organization’s long-term goals. A recognizable cost variance could be an increase in repair costs as a percentage of sales on an increasing basis. This variance could indicate that equipment is not operating efficiently and is increasing overall cost.