Back to the on-screen lesson ·
Use totals, ordered data and transformations to reason about statistical summaries.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Compute and interpret descriptive summaries while preserving group weights, data multiplicity and stated quartile conventions.
A data set can contain repeated values. The list 2, 2, 2, 8 contains four observations, not two, and its mean is fourteen divided by four. A frequency table compresses repetitions but does not remove them. Sort values before finding a median or quartile, keeping each occurrence. If the data are grouped into intervals, an exact mean or median may not be recoverable without additional assumptions about positions inside those intervals.
| Term | What it means |
|---|---|
| Mean | Sum of the observations divided by their number. |
| Median | The middle sorted observation, or mean of the two middle observations for an even count. |
| Interquartile range | The difference Q3 minus Q1, describing the span of the middle half. |
| Standard deviation | A nonnegative measure of how far observations typically lie from their mean. |
The mean describes a balance point, the median describes the middle ordered position, and the mode identifies the most frequent value. The range and standard deviation describe different aspects of spread. Quartiles and percentiles describe relative positions. These measures are related but not interchangeable. A mean does not reveal every observation, and equal means do not imply equal distributions.
Begin by identifying what information a summary preserves. A mean together with a count gives a total, which can be combined with another total. A median tells you about ordered positions but usually cannot reconstruct a sum. A standard deviation describes distances from the mean, not the location of the mean itself. Choosing the correct representation often makes a problem short without requiring a lengthy formula calculation.
Another way: steps
Identify the summary and data weights; reconstruct totals or ordered positions; calculate; interpret only what the summary supports.
If n values have mean m, their total is nm. Adding a new observation x changes the mean to (nm + x)/(n + 1). Removing a known observation subtracts it from the total and reduces the count by one. This approach prevents the error of averaging the old mean with the new observation as though each represented one value. It also solves missing-score problems: find the required total, subtract the sum of known scores, and check that the resulting missing value respects any stated score bounds.
A class of ten with mean eighty contributes total eight hundred, while a class of thirty with mean sixty contributes eighteen hundred. The combined mean is twenty-six hundred divided by forty, or sixty-five. The unweighted average of the two means is seventy and gives the small class too much influence. Weight by the number of observations each mean represents. When weights are proportions adding to one, multiply each mean by its proportion and add. A combined mean must lie between the group means, but that bound alone does not verify the correct weighting.
For an odd number of ordered observations, take the single middle value. For an even number, average the two central values. The median need not appear among the observations. The mode is a most frequent value; data can have more than one mode, and conventions may describe a set with no repeated values as having no mode. If a question depends on that distinction, use its stated convention. A large outlier can shift the mean substantially while leaving the median unchanged, because changing magnitude need not change the middle position.
Range is maximum minus minimum, so it uses only the endpoints. Two distributions with the same range can place their internal observations very differently. Quartiles divide ordered data into broad portions, and the interquartile range is Q3 - Q1. Several computational quartile conventions exist for small samples. These questions state the convention when calculating from a short list; do not assume a software package's convention is universal. When Q1 and Q3 are already supplied in a boxplot, their difference needs no reconstruction of the individual observations. Read a boxplot legend before interpreting its whiskers: some displays use the minimum and maximum, while others mark outliers separately. The box still identifies the quartile interval, but a whisker endpoint need not identify the overall extreme under every plotting convention.
If every observation increases by c, the mean also increases by c and every deviation from the mean stays the same. Standard deviation therefore stays unchanged. If every observation is multiplied by k, each distance from the mean multiplies by |k|, so standard deviation does too. The absolute value matters for a negative multiplier because spread cannot be negative. A distribution with all observations equal has standard deviation zero. Comparing spreads often requires no square roots: values clustered tightly about the same mean have smaller standard deviation than values shifted farther from it in balanced pairs.
A population variance averages squared deviations using divisor N. The usual sample variance estimator uses divisor n - 1. A question requiring an exact computed variance or standard deviation must make its convention clear. Do not silently swap formulas. Many GRE-style reasoning questions instead use properties that hold for either convention, such as invariance under a common shift and scaling by the absolute multiplier. If the task only asks which distribution is more spread out, an exact formula may add work without improving the reasoning.
A score at the eightieth percentile describes its position relative to a comparison group; it does not mean the person answered eighty percent of questions correctly. Ties and percentile definitions can affect exact counts, so be cautious when converting a rank statement into a precise number below a score. If the question supplies a definition such as the percent of observations strictly below a value, use that definition. A percentile can change when the comparison population changes even if the person's raw measurement remains fixed.
A report combines two survey groups with different numbers of responses. To verify its reported overall mean, reconstruct each group's total rating and divide the combined total by the combined response count. An unweighted average of the two group means gives the smaller group too much influence. The correction is mathematical even if both group means were individually accurate.
Then ask what the corrected mean still does not show. It does not establish that every respondent felt similarly, that the sample represents the population, or that a change in the mean was caused by a particular intervention. A distribution with half very low and half very high ratings can share a mean with one concentrated near the middle. Additional summaries or the underlying distribution are needed for those questions.
In this course, automatically graded items test bounded calculations and interpretations. Keep broader claims about representativeness or causation separate from arithmetic correctness. A useful error record distinguishes a wrong weight, an ordering mistake, a unit error and an unsupported inference beyond the supplied summaries.
Do not average group means without checking group sizes. Sort data before finding a median, and retain repeated observations. A percentile is a relative position, not a percentage score. Adding a constant changes location but not spread. Multiplying by a negative number reverses order but cannot produce a negative standard deviation. Use the stated quartile convention for small samples.
Recover the required total.
Six observations have mean 14, so their sum is 84.
The mean represents all six values.
Add the five known observations.
8 + 11 + 13 + 16 + 19 = 67.
Each listed value counts once.
Find the remaining contribution.
84 - 67 = 17.
The missing observation must supply the difference.
Verify the mean.
(67 + 17)/6 = 14.
The reconstructed total satisfies the original statement.
Identify the group weights.
Nine values have mean 20; six values have mean 35.
The groups have different counts.
Recover each group's total.
9 times 20 = 180; 6 times 35 = 210.
Totals, unlike group means, can be added directly.
Combine totals and counts.
Total 390 across 15 observations.
Every observation has equal weight in the combined mean.
Compute the combined mean.
390/15 = 26.
The result is closer to twenty because that group is larger.
Check a tempting alternative.
The unweighted mean 27.5 is incorrect.
It would give both groups equal influence despite unequal counts.
Read the original summaries.
A data set has mean 12, range 10 and standard deviation 3.
Every value is replaced by -2x + 7.
Transform the mean.
New mean = -2(12) + 7 = -17.
The mean follows an affine transformation of every observation.
Separate scale from shift.
The multiplier is -2 and the shift is 7.
The shift changes location but not distances.
Transform the range.
New range = 2 times 10 = 20.
A negative multiplier reverses order, but range stays nonnegative.
Transform standard deviation.
New standard deviation = 2 times 3 = 6.
All distances from the mean double.
Check the interpretation.
The mean can be negative; the two spread measures cannot.
Do not carry the multiplier's negative sign into a distance measure.
Use the stated convention.
Data: 1, 2, 5, 7, 9, 10, 13, 15; use medians of the lower and upper halves.
There are four observations in each half.
Locate each half's middle pair.
Q1 = (2+5)/2; Q3 = (10+13)/2.
Compute their difference after finding both quartiles.
Subtract in the correct order.
What is the mean of 4, 6, 9 and 13?
Answer:
What is the median of 8, 2, 11, 5 and 7?
Answer:
Five numbers have mean 18. Four of the numbers are 12, 16, 19 and 23. What is the fifth number?
Answer:
A group of 8 scores has mean 70 and a group of 12 scores has mean 80. What is the mean of all 20 scores?
Answer:
What is the median of 3, 4, 4, 9, 12 and 16?
Answer:
A data set has minimum 7 and maximum 31. Every observation is replaced by twice its value minus 5. What is the new range?
Answer:
A distribution has standard deviation 6. Each observation is multiplied by -3 and then increased by 10. What is the new standard deviation?
Answer:
For the sorted data 2, 4, 6, 8, 10, 12, 14, 16, define Q1 and Q3 as the medians of the lower and upper halves. What is the interquartile range?
Answer:
Scores of 10, 20 and 30 occur with frequencies 2, 5 and 3 respectively. What is the mean score?
Answer:
A set of 7 observations has mean 15. A new observation of 31 is added. What is the new mean?
Answer:
A report combines 30 survey responses with mean rating 4.2 and 20 responses with mean rating 3.2. What is the combined mean rating?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Four observations have mean 22. One observation of 10 is removed. What is the mean of the remaining three?
Answer:
After a study gap, reconstruct a weighted mean from group totals and explain which summaries change when every value increases by the same constant. Review introductory statistics for more work with distributions and graphs.
17. Complete a quartile calculation, step 3
Q1 = 3.5; Q3 = 11.5; IQR = 8.
The interquartile range measures spread and is nonnegative.