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Equal standard deviations mean that two datasets have the same value of one spread measure, using the same definition and compatible units. They do not mean that the datasets contain the same observations, have the same mean, share the same range, or follow the same pattern. A summary describes a feature of data; it does not reconstruct the full list.
For G3 Mathematics, compare the mean and standard deviation separately, name the measured quantity, and keep the conclusion within what those summaries support. If two groups have equal standard deviations, you can report equal spread by that measure. You cannot conclude that every score, time or measurement is equally distributed across the two groups without further information.
Contents
Sections 1–5
Sections 6–10
Sections 11–15
Sections 16–20
Sections 21–22
Consider Dataset A: 2, 2, 5, 5. Dataset B: 2, 3, 3, 6. Both have mean 3.5. Using the standard deviation defined with division by the number of observations, both have standard deviation 1.5. Yet A has range 3 and median 3.5, while B has range 4 and median 3.
This pair makes the limitation visible. Even knowing both the mean and standard deviation does not uniquely identify every value. The datasets agree on two summaries and disagree on other features.
There is an even simpler way for equal standard deviations to occur: add the same constant to every value in a dataset. The mean moves by that constant, but each observation’s distance from its own mean stays the same. Standard deviation is unchanged while the whole dataset shifts.
This article focuses on interpreting equality of standard deviations. Use the standard-deviation mastery guide for the wider topic of calculation and spread. Return here when a comparison sentence seems to claim more than one spread statistic can establish.
The SEAB G3 Mathematics syllabus for 2027 includes standard deviation for grouped and ungrouped data and using the mean and standard deviation to compare datasets. It is therefore important to know both what the calculation measures and what a comparison based on it can reasonably say.
The examples here use small invented lists so every step can be inspected. They are not claims about actual classes, schools or examination results. Their purpose is to separate a numerical summary from the observations it summarises.
Unless stated otherwise, this guide uses the descriptive standard deviation obtained by dividing the sum of squared deviations by n, the number of observations, before taking the square root. Some calculators also display a sample standard deviation using n − 1. Choose the output required by the question rather than assuming that all displayed standard deviations use the same definition.
This distinction matters in comparisons. Two numerical outputs are meaningful together only when their definitions, units and contexts are compatible. A calculation can be arithmetically correct and still support a misleading comparison if one output describes seconds and the other minutes, or one uses a different convention.
For the broader subject route, see what G1, G2 and G3 Mathematics mean. Learn the interpretation alongside the calculator procedure; neither needs to wait until the other has become a memorised routine.
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Standard deviation measures spread relative to the dataset’s own mean. Calculate the deviations from that mean, square them, find their mean under the stated convention, and take the square root. Squaring prevents positive and negative deviations from cancelling in the total.
For 2, 2, 5, 5, the mean is 3.5. The deviations are −1.5, −1.5, 1.5 and 1.5. Their squares are all 2.25. The mean squared deviation is 2.25, so the standard deviation is the square root of 2.25, which is 1.5.
The statistic combines information about all the observations into one number. That compression is useful because it makes comparisons manageable. It also means that different patterns can produce the same result.
Do not describe standard deviation as the ordinary average of the absolute distances from the mean. That is a different calculation. Standard deviation uses squared deviations before the square root. The sequence of operations is part of the definition.
The unit of standard deviation matches the unit of the original observations. If the data are times in minutes, the standard deviation is in minutes. The squared deviations and variance have squared units during the calculation, but the square root returns the final spread measure to the original unit. This makes a contextual comparison easier to express precisely.
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Start with Dataset A: 2, 2, 5, 5. Its mean is 3.5 and its standard deviation is 1.5. Add 10 to every observation to obtain Dataset C: 12, 12, 15, 15. C’s mean is 13.5.
The deviations from C’s own mean are again −1.5, −1.5, 1.5 and 1.5. Therefore its standard deviation is also 1.5. The whole dataset has shifted without changing its internal spread about its mean.
If these were invented task times in minutes, C would have a higher mean completion time while the standard deviations match. Saying that the groups performed identically would ignore the ten-minute difference in their means.
The comparison should state the two features separately: “C has a mean ten minutes higher, while both groups have standard deviation 1.5 minutes.” If lower completion time is relevant to the question, explain that context without treating the spread statistic as a measure of speed.
This example shows why standard deviation is not a measure of the overall level of the data. It measures deviations from each group’s own centre. Moving the centre and moving the spread are different operations, so the interpretation should not merge them into one judgement.
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SECTION 5 OF 22
4. Equal means and equal standard deviations can still hide different data
Now compare A: 2, 2, 5, 5 with B: 2, 3, 3, 6. Both totals are 14, so both means are 14/4 = 3.5. Their observations differ, but the mean alone does not show how.
For B, the deviations from 3.5 are −1.5, −0.5, −0.5 and 2.5. Their squares are 2.25, 0.25, 0.25 and 6.25. The squared deviations sum to 9, giving mean squared deviation 9/4 = 2.25 and standard deviation 1.5.
A also has squared-deviation sum 9. Thus both mean and standard deviation match. However, A’s median is (2 + 5)/2 = 3.5, while B’s median is (3 + 3)/2 = 3. Their ranges are 5 − 2 = 3 and 6 − 2 = 4.
| Feature | Dataset A: 2, 2, 5, 5 | Dataset B: 2, 3, 3, 6 |
|---|---|---|
| Standard deviation using n | 1.5 | 1.5 |
| Median | 3.5 | 3 |
| Range | 3 | 4 |
The pair demonstrates that these two summaries do not determine the full dataset. It does not make the summaries wrong or useless. It shows the boundary of the information they preserve.
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Dataset A has four equal squared deviations of 2.25. Dataset B has one larger squared deviation of 6.25, one of 2.25, and two small ones of 0.25. Their totals match despite different arrangements of individual distances from the mean.
This is the mechanism behind the equal standard deviations. The calculation depends on the total squared deviation and the number of observations, not on a unique pattern of individual deviations. Many different combinations can produce the same total.
A learner may expect the larger maximum in B to force a larger standard deviation. The maximum contributes to spread, but other observations contribute too. In B, the two values of 3 lie close to the mean, offsetting the effect of the value 6 within the sum of squares.
Do not turn this explanation into a rule that extremes never matter. They matter strongly because deviations are squared. The point is that one extreme alone does not determine the whole sum; the remaining observations and the mean must also be considered.
Inspecting the deviations is a useful teaching step even if the final examination calculation uses a calculator or the formula involving sums of squares. It reveals how equal numerical outputs can emerge from different lists and prevents the statistic from becoming a mysterious button result.
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Range uses only the maximum and minimum. Standard deviation uses every observation through its deviation from the mean. Equal standard deviations therefore do not force equal ranges, as A and B show.
Conversely, equal ranges do not force equal standard deviations. Compare 0, 0, 10, 10 with 0, 5, 5, 10. Both have range 10 and mean 5. The first has standard deviation 5. The second has squared deviations 25, 0, 0 and 25, giving standard deviation square root of 12.5, approximately 3.54.
The endpoints match, but the middle observations differ. The second list places more observations at the mean, reducing the mean squared deviation. Range does not register that change because it ignores the interior values.
If a question asks for a comparison using standard deviation, use that measure and explain it. You may mention another statistic if it is relevant and available, but do not substitute range simply because it is easier to calculate.
The cumulative-frequency and box-plot mastery guide develops other ways to describe position and spread. Different measures can complement one another because each preserves different features of a distribution. They should not be treated as interchangeable names for one universal notion of consistency.
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In school comparisons, a smaller standard deviation is often described as greater consistency about the mean. That can be useful shorthand when the measured quantity, unit and definition are clear. But “consistent” should not become a claim that every observation is close to every other observation or that no unusual value exists.
If two groups have equal standard deviations, say their spread about their respective means is equal by that measure. If the values are rounded, say the reported standard deviations are equal at the stated precision. These are narrower and more accurate claims than “the groups are exactly the same.”
Context determines whether lower spread is desirable. For a process intended to produce a fixed measurement, lower variability may be useful. For a collection intended to contain diverse values, uniformity may not be the goal. A mathematics question may specify the interpretation it wants.
Do not infer effort, ability or teaching quality from a standard deviation alone. The statistic summarises the observed values; it does not identify the causes of their distribution. Additional evidence would be needed for those explanations.
A careful comparison sentence can be short: “The groups have equal reported standard deviations, so neither has smaller spread by this measure; their means still need to be compared separately.” This gives the reader a clear conclusion while respecting what the numbers do and do not establish.
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Let each observation increase by a constant k. The mean also increases by k. A deviation becomes (x + k) − (mean + k), which simplifies to x − mean. Every deviation is unchanged, so every squared deviation and the standard deviation remain unchanged.
This reasoning covers adding 10 in the earlier example, but it applies to any constant shift in the same numerical scale. Subtracting a constant works similarly. A shift changes the centre without changing distances between observations.
For an invented dataset measured in points, adding a fixed bonus of 4 points to every value increases the mean by 4 and leaves the standard deviation unchanged, provided no cap or other rule alters the transformation. If scores are capped, the operation is no longer the same constant shift for every observation.
That condition matters. A rule such as “add 4, but stop at the maximum score” can compress high values differently from low ones. You cannot apply the simple unchanged-standard-deviation rule without checking whether all observations actually received the same addition.
This is a useful habit in transformation questions: state the operation precisely, then use the corresponding statistical effect. Everyday descriptions such as bonus, adjustment or correction do not by themselves guarantee that the mathematical transformation is a uniform shift.
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If every observation is multiplied by a positive constant a, the mean is multiplied by a and every deviation from the new mean is multiplied by a. The squared deviations are multiplied by a². Taking the square root means the standard deviation is multiplied by a.
For Dataset A, multiplying every value by 2 gives 4, 4, 10, 10. The mean becomes 7 and the standard deviation becomes 3. Both are twice the original values of those summaries.
If the multiplier is negative, standard deviation is multiplied by its absolute value because a spread measure is nonnegative. Multiplying by −2 reflects the numerical positions and scales their distances by 2. The mean changes sign and scale, while the standard deviation becomes 3 rather than −3.
The combined transformation y = ax + b therefore changes standard deviation by the factor |a| and shifts the mean according to a times the old mean plus b. Use this relationship only when the same transformation applies to every observation.
A special case is a = 0. Every transformed observation equals b, so the standard deviation is zero. The old spread disappears because all values have been mapped to the same number. This is another reminder that the transformation, rather than a vague description of adjustment, controls the result.
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Suppose one group has standard deviation 2 minutes and another has standard deviation 90 seconds for the same type of measurement. Comparing the bare numbers 2 and 90 would be misleading. Convert 2 minutes to 120 seconds; then the second group has the smaller standard deviation.
Unit conversion is a scaling transformation. Multiplying every time in minutes by 60 multiplies the standard deviation by 60. Thus 2 minutes corresponds to 120 seconds, consistent with the meaning of the statistic.
The mean must be converted too if you compare centres. Keep each summary attached to its unit throughout the working. A table with columns for mean and standard deviation can become misleading if the units are assumed rather than written.
Equal numerical standard deviations in different units do not establish equal spread. A value of 1 metre and a value of 1 centimetre describe different distances. The unit is part of the quantity, not a decorative label added after the comparison.
The guide to reading quantities before calculating develops this foundational check. It remains important in G3 because a sophisticated calculation cannot repair a comparison that began with incompatible measurement scales.
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Suppose two standard deviations are 1.46 and 1.54 units. Rounded to one decimal place, both are reported as 1.5. Their reported values match, but the unrounded values differ.
If a question supplies only those rounded summaries, you should not claim exact equality of the underlying standard deviations. Say that the reported standard deviations are equal at the given precision. If the original data are available, retain enough precision during calculation to make the requested comparison.
This is different from the earlier A and B example, where the exact descriptive standard deviations are both 1.5. There, equality is genuine within the model, yet the datasets differ. Rounded equality introduces an additional limitation on top of the information already lost through summarising data.
Do not invent a meaningful performance difference from a tiny numerical gap without considering the question and the evidence. A school exercise may simply ask which calculated spread is larger. Broader claims about real groups require more than a last-decimal comparison.
The guide to exact values and graph estimates explains why the source and precision of a number affect the claims it supports. Apply that discipline to calculator outputs too: a displayed value is a representation of a quantity, with a precision that should remain visible in the interpretation.
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Many calculators distinguish a descriptive standard deviation using n from a sample standard deviation using n − 1. The two outputs are related but are not the same number for a nonconstant finite list. Choose the one the question requires.
For a squared-deviation sum of 9 across four observations, division by n gives variance 9/4 and standard deviation 1.5. Division by n − 1 gives variance 9/3 = 3 and standard deviation square root of 3, approximately 1.732.
If one learner reports 1.5 and another 1.732 for the same list, inspect their selected outputs before assuming that someone entered the data incorrectly. The difference may come from convention rather than arithmetic.
For our A and B lists, both have the same n and squared-deviation sum, so they also match under the same alternative sample convention. That does not make the convention irrelevant; it shows that the comparison should consistently use one definition.
Be especially careful when group sizes differ. The relationship between the two conventions depends on n, so mixing definitions can distort a comparison. Read the question, inspect the calculator labels, and record the output chosen. A reliable routine includes selecting the intended statistic rather than treating every symbol containing an s or a sigma as interchangeable.
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A frequency table can represent the same dataset more compactly. Dataset A can be recorded as value 2 with frequency 2 and value 5 with frequency 2. Dataset B can be recorded as values 2, 3 and 6 with frequencies 1, 2 and 1.
When calculating the mean or standard deviation, the frequencies matter because they say how many observations contribute each value. Treating each distinct row as one observation would change both the total count and the distribution.
For A, the total count is 4 and the weighted sum is 2 × 2 + 5 × 2 = 14. The weighted sum of squares is 2² × 2 + 5² × 2 = 58. Thus the variance is 58/4 − 3.5² = 2.25.
For B, the weighted sum is 2 × 1 + 3 × 2 + 6 × 1 = 14. The weighted sum of squares is 4 + 18 + 36 = 58. The same count, sum and sum of squares produce the same mean and standard deviation despite different frequencies at different values.
This representation shows exactly what these summaries retain. It also shows what they do not retain: the frequency pattern itself is not uniquely recoverable from those three aggregate quantities. Keep the original table when the question asks about modes, particular categories or threshold counts.
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SECTION 15 OF 22
14. A threshold comparison needs more than mean and standard deviation
Take A: 2, 2, 5, 5 and B: 2, 3, 3, 6. Both have mean 3.5 and standard deviation 1.5. At a threshold of at least 5, A has two qualifying observations out of four, while B has one out of four.
Thus equal means and standard deviations do not force equal proportions meeting a particular threshold. A has 50% at least 5, while B has 25%. The actual distribution around that threshold matters.
Do not infer these proportions from a bell-shaped picture unless a suitable distributional model is explicitly given and relevant. The small lists here are fully specified and do not require any assumed shape. A mean and standard deviation alone do not establish that data follow a particular curve.
For cumulative-frequency work, the guide to comparing groups with different totals explains how to convert counts to shares using each group’s denominator. Here the totals match, so the count-to-percentage step is simple, but the missing distributional information remains important.
This is a useful test of an overstrong summary claim. If someone says that equal mean and spread imply equal outcomes at every threshold, the two lists provide a concrete disagreement. The correct response is to request the relevant distribution or counts, rather than asking the summaries to contain information they have discarded.
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Question A: Find the mean and descriptive standard deviation of 2, 2, 5, 5. Then find the same summaries for 12, 12, 15, 15. Explain why the standard deviations match despite different means.
Question B: Verify that 2, 3, 3, 6 has the same mean and descriptive standard deviation as 2, 2, 5, 5. Compare their medians and ranges. State one conclusion that would be too strong if based only on their matching summaries.
Question C: Every value in a dataset is multiplied by 3 and then increased by 7. The original mean is 4 and standard deviation is 2. Find the transformed mean and standard deviation, assuming the same operation applies to every observation.
Question D: One group has standard deviation 1.5 minutes and another 100 seconds for comparable times. Which has the smaller spread by this measure? Convert to a common unit before comparing.
Question E: Two reported standard deviations are both 2.4 to one decimal place. Must the exact values be equal? Give two possible exact values that would produce those reports under ordinary rounding.
Question F: For A: 2, 2, 5, 5 and B: 2, 3, 3, 6, compare the percentages at least 5. Explain why the common mean and standard deviation do not settle this threshold comparison.
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For A, the first mean is 3.5 and the second is 13.5. Both descriptive standard deviations are 1.5. Adding 10 changes each observation and its mean by the same amount, leaving every deviation unchanged.
For B, both means are 3.5 and both squared-deviation sums are 9, so both standard deviations are 1.5. The medians are 3.5 and 3, while the ranges are 3 and 4. “The datasets are identical” would be too strong.
For C, the transformed mean is 3 × 4 + 7 = 19. The standard deviation is 3 × 2 = 6. The constant addition shifts the centre but does not add 7 to the spread.
For D, 1.5 minutes equal 90 seconds. The first group therefore has the smaller standard deviation, 90 seconds compared with 100 seconds. Comparing the bare numbers 1.5 and 100 would not show that the units had been understood.
For E, exact values need not match. For example, 2.36 and 2.44 both round to 2.4 at one decimal place. The equality is at the reported precision.
For F, A has 2/4 = 50% at least 5, while B has 1/4 = 25%. The summaries do not uniquely specify how many observations fall on either side of that threshold. The full lists provide the needed information.
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A learner writes, “Both standard deviations are 1.5, so both groups got the same results.” Ask which part of the sentence follows from the numbers. Equal spread by the stated measure follows; identical results do not. Replace only the unsupported clause.
Another learner writes, “The larger standard deviation means the mean is higher.” Use a constant-shift example to separate centre from spread. Two groups can have different means and equal standard deviations, so neither statistic automatically determines the other.
A third learner compares 2 minutes with 90 seconds without conversion. The repair is a unit check, not a new standard-deviation calculation. A fourth selects the n − 1 output when the question uses n. The repair is identifying the intended statistic.
The marked-paper guide helps turn each of these errors into a specific next action. The category statistics is too broad to identify the repair by itself.
When teaching, ask for a sentence containing the measured quantity, the unit and the named statistic. “The completion times have equal reported standard deviations of 1.5 minutes” is clearer than “they are the same.” Precision in language is part of precision in mathematics.
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SECTION 19 OF 22
18. A useful extension: identical relative shape is still a separate claim
When a dataset is shifted by a constant, its pattern of deviations remains exactly the same. That is a stronger relationship than merely having an equal standard deviation. Our datasets A and C are shifts of one another, so the correspondence can be shown observation by observation.
Datasets A and B have equal standard deviations but are not constant shifts of one another. Their means already match, yet their sorted values differ. The equality of one spread statistic does not supply the stronger correspondence found in a known translation.
This distinction helps with reasoning from transformations. If the problem tells you that every observation was increased by 10, you know a specific mechanism and can derive its consequences. If it tells you only that standard deviations match, you cannot reverse the logic and conclude that a constant was added.
Many mathematical relationships work in this direction: a known operation guarantees a particular summary effect, but that effect can arise through other operations too. Preserve the direction of the implication.
For a broader explanation of constructing alternative possibilities, use the two-valid-examples guide. It explains why two compatible cases with different target features can demonstrate that the supplied information does not determine that feature uniquely.
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SECTION 20 OF 22
19. Repeating every observation can preserve the descriptive spread
Take Dataset A: 2, 2, 5, 5. Repeat every observation twice to obtain 2, 2, 2, 2, 5, 5, 5, 5. The total count doubles from 4 to 8, and the total sum doubles from 14 to 28. The mean remains 3.5.
The squared-deviation sum also doubles, from 9 to 18. Dividing by the doubled count gives the same descriptive variance: 18/8 = 2.25. Therefore the descriptive standard deviation remains 1.5.
This is another way equal standard deviations can occur without identical datasets. The frequency pattern has been scaled proportionally. The new list contains more observations but has the same relative distribution across the two values.
Do not infer that a larger dataset must have a larger standard deviation merely because more squared deviations are added. The descriptive calculation divides that total by the number of observations. Both the numerator and denominator change under uniform repetition.
Also do not transfer this exact unchanged result to a different standard-deviation convention without checking. Under division by n − 1, the original list gives square root of 3, while the repeated list gives square root of 18/7. Those are different because the denominator does not simply double.
The example reinforces why the definition belongs in the comparison. “Repeat all observations” has a clear effect on the descriptive distribution, but a calculator output with another convention may change. Name the statistic before using a transformation rule.
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An invented table gives Group P mean 42 units and standard deviation 6 units. Group Q has mean 47 units and standard deviation 6 units. Both summaries use the same descriptive definition and measure comparable quantities on the same scale.
The supported comparison is that Q has the higher mean by 5 units, while the reported spreads about the respective means are equal by standard deviation. The table does not show that Q is P shifted upward by exactly 5 at every observation. A uniform shift would produce those summaries, but other distributions could produce them too.
If the table then gives Group R mean 42 and standard deviation 9, P and R share a mean but differ in spread. R has the larger standard deviation. This does not tell you the exact range, median or number above a threshold for either group.
Ask what additional information would answer a threshold question. A frequency table, raw list or suitable cumulative distribution could provide the relevant counts. Another single summary might still leave several possible distributions. Choose information that addresses the target instead of merely collecting more statistics.
This makes a useful short exercise in evidence discipline. For each sentence, underline the statistic that supports it. If a sentence claims an exact observation pattern, a causal explanation or a threshold count that no supplied statistic determines, mark it as unsupported and rewrite it at the appropriate scope.
A good comparison is neither vague nor overstated. It names the quantity and unit, separates centre from spread, and acknowledges the precision of the reports. That is enough to make a useful conclusion without pretending that a two-number summary is a complete picture of every member of the group.
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Does equal standard deviation mean equal variance?
When both use the same definition and compatible units, equal nonnegative standard deviations have equal squares, so the corresponding variances are equal. That still does not determine the individual observations.
Can standard deviation be zero for different datasets?
Yes. Every observation in 4, 4, 4 is equal to its mean, so its standard deviation is zero. The same is true for 9, 9, 9. Each list has no internal variation, but their levels differ. Zero spread does not identify the common value without more information.
Should I always calculate more statistics?
Calculate what the question requires and what the comparison needs. The lesson is to respect the scope of the available evidence, not to produce every possible summary for every dataset. Sometimes the right conclusion is simply that a requested feature cannot be inferred from the supplied summaries.
Can I say one group is more consistent?
Use the named measure and the context. A smaller standard deviation supports a statement about smaller spread about the mean by that measure. Equal reported standard deviations do not support saying that one has smaller spread on the available precision alone.
Where should I go next?
Use the Mathematics explanation directory for another focused question. The Clementi Secondary 1 Mathematics small-group guide gives the local teaching context. The immediate habit is to read a standard deviation as one precise summary of spread, keep the mean separate, and ask for the original distribution when the conclusion needs information that the summaries do not preserve.
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