Free math worksheets

- Identify the whisker endpoints, box edges, and median line separately. - Record those five features in their left-to-right order.

1. The left whisker is the minimum \(4\). 2. The left box edge is \(Q_1=7\), the center line is the median \(11\), and the right box edge is \(Q_3=15\). 3. The right whisker is the maximum \(19\).

\((4,7,11,15,19)\).

- Use the whisker endpoints for one spread and the box endpoints for the other. - Range uses maximum minus minimum; IQR uses \(Q_3-Q_1\).

1. The range is \(33-10=23\). 2. The IQR is \(25-16=9\).

Range: \(23\). IQR: \(9\).

- Use the four quartile sections represented by the boxplot. - Each quartile interval represents about one-fourth of the ordered observations regardless of its width.

1. The box from \(Q_1\) to \(Q_3\) contains the middle \(50\%\). 2. The interval from the median to the maximum contains the upper half, approximately \(50\%\).

About \(50\%\) lie between \(20\) and \(36\), and about \(50\%\) lie from the median to the maximum.

- Use the relationship between the two box edges and the box width. - Rearrange \(\text{IQR}=Q_3-Q_1\) to isolate the unknown quartile.

1. The IQR is \(Q_3-Q_1\). 2. \(Q_3=18+11=29\).

\(Q_3=29\).

- Work backward from the definition of range. - Rearrange \(\text{range}=\text{maximum}-\text{minimum}\) to solve for the minimum.

1. The range equals maximum minus minimum. 2. The minimum is \(84-39=45\).

The minimum is \(45\).

- Recall the required order from minimum through maximum. - Reject any tuple in which a quartile or the median falls outside its neighboring landmarks.

1. A five-number summary must be nondecreasing from minimum through maximum. 2. Only \((5,10,12,18,22)\) satisfies the required order.

b) \((5,10,12,18,22)\).

- Match the percentile to its quartile landmark. - The \(75\)th percentile is the right edge of the box; interpret it as an approximate cumulative proportion.

1. The \(75\)th percentile is the third quartile. 2. The right edge of the box gives \(Q_3=31\). 3. Approximately \(75\%\) of observations are at or below \(31\).

The \(75\)th percentile is \(31\); approximately \(75\%\) of observations are at or below it.

- List the quantities represented directly by a boxplot. - Ask whether each observation receives its own visible mark. - Consider what information is discarded when data are reduced to five summary values.

1. A boxplot displays selected summary values and possibly outliers, not one mark for every observation. 2. Samples with different numbers of observations can share the same minimum, quartiles, median, maximum, and outlier pattern. 3. Therefore the sample size cannot be recovered unless it is stated separately.

No. A boxplot does not encode sample size, and samples of different sizes can have the same five-number summary.

- Separate visual layout from numerical encoding. - Track the marked summary values rather than their page direction. - Rotation preserves distances along the value axis.

1. Changing orientation changes only the layout of the display. 2. The minimum, quartiles, median, maximum, outliers, and all numerical segment lengths remain unchanged. 3. No conclusion about center, spread, or skewness should change solely because the plot is rotated.

No statistical information changes. Only the orientation of the display changes.

- Order the values and find the median of each half. - For an even sample, average the two middle values for the overall median and for each half.

1. The minimum and maximum are \(2\) and \(14\). 2. The median is \(\frac{7+8}{2}=7.5\). 3. The lower-half median is \(Q_1=4.5\), and the upper-half median is \(Q_3=11\).

The five-number summary is \((2,4.5,7.5,11,14)\).

- Distinguish symbols outside the whiskers from the whisker endpoints. - When outliers are plotted separately, whiskers stop at the most extreme nonoutlying observations.

1. The whiskers end at \(21\) and \(38\). 2. The plotted outliers are \(12\) and \(49\). 3. With outliers shown separately, whisker endpoints are the most extreme nonoutliers, not necessarily the data extrema.

Whiskers: \(21\) and \(38\). Outliers: \(12\) and \(49\). The whiskers are nonoutlier endpoints, not the full-data minimum and maximum.

- Compare the two whisker lengths and the median’s position inside the box. - A longer right side suggests a high-value tail; the mean tends to be pulled toward that tail.

1. The right whisker is much longer than the left whisker, suggesting a long right tail. 2. The distribution is likely skewed right. 3. For a right-skewed distribution, the mean is usually greater than the median.

The distribution is likely right-skewed, so the mean is likely greater than the median.

- Look for the longer tail and connect its direction to the mean. - A longer left side suggests a low-value tail; compare how that tail affects the mean.

1. The left whisker is much longer, suggesting a long left tail. 2. The distribution is likely skewed left. 3. The mean is usually smaller than the median in a left-skewed distribution.

The distribution is likely left-skewed, and the mean is likely smaller than the median.

- Separate a positional summary from an exact-value frequency. - The median describes position, not the number of observations equal to its numerical value.

1. The median gives a central ordered position or average of two central positions. 2. A boxplot does not show the frequency of any exact value. 3. There may be zero, one, or multiple observations equal to \(14\).

The boxplot shows the median value but not how many observations equal it.

- Recall the proportion represented by each adjacent quartile interval. - Use interval width to describe numerical spread, not the number of observations.

1. Each adjacent quartile interval contains approximately \(25\%\) of the observations. 2. The wider interval indicates that its observations are more spread out, not that it contains more observations.

No. Each interval contains about \(25\%\) of the data; the median-to-\(Q_3\) values are simply more spread out.

- Apply the increasing transformation to every landmark. - Check all five transformed values and preserve their order.

1. Apply the increasing linear transformation to each summary value. 2. The transformed values are \(2(3)+4=10\), \(2(5)+4=14\), \(2(8)+4=20\), \(2(11)+4=26\), and \(2(15)+4=34\).

The transformed five-number summary is \((10,14,20,26,34)\).

- Use the same unit conversion on every landmark. - Check that the converted summary remains increasing and is labeled in inches.

1. Divide each value by \(2.54\). 2. The converted values are \(10,20,30,40,50\) inches.

\((10,20,30,40,50)\) inches.

- Compare corresponding distances on the left and right sides. - Check whether the median bisects the box and whether the two whiskers have similar lengths.

1. The median is centered in the box because \(10-6=14-10=4\). 2. The whiskers have equal lengths because \(6-2=18-14=4\). 3. These relationships suggest approximate symmetry.

Yes. The median is centered in the box and the whiskers have equal lengths.

- Compare the mean marker with the median line. - Use the longer whisker or side of the box as supporting evidence for the skew direction.

1. The mean marker is at \(12\), while the median is \(9\). 2. The mean is greater than the median. 3. Together with the longer right side, this suggests right skew.

Mean \(=12\), median \(=9\); the distribution is likely right-skewed.

- Consider how much of the original data the representation retains. - A boxplot omits the exact values and frequencies within each quartile interval.

1. A boxplot records only the five-number summary and any displayed outliers. 2. Many different arrangements of values within the quartile intervals can share those same landmarks. 3. Therefore, distinct raw data sets can produce the same boxplot.

Yes. A boxplot does not preserve all individual values, so different data sets can share the same five-number summary and boxplot.

- Locate the two central positions of the full ordered list. - Treat the first six and last six observations as separate ordered halves. - Each even-sized list has a center halfway between two positions.

1. The minimum and maximum are \(x_1\) and \(x_{12}\). 2. The lower half is \(x_1\) through \(x_6\), so \(Q_1=\frac{x_3+x_4}{2}\). 3. The overall median is \(\frac{x_6+x_7}{2}\). 4. The upper half is \(x_7\) through \(x_{12}\), so \(Q_3=\frac{x_9+x_{10}}{2}\).

Minimum: \(x_1\) \(Q_1=\frac{x_3+x_4}{2}\) Median: \(\frac{x_6+x_7}{2}\) \(Q_3=\frac{x_9+x_{10}}{2}\) Maximum: \(x_{12}\)

- Even-sized ordered lists have centers between two observations. - Apply the same idea to each four-observation half. - A summary statistic need not equal a recorded observation.

1. The minimum and maximum are \(1\) and \(8\). 2. The median is \(\frac{4+5}{2}=4.5\). 3. The lower-half median is \(Q_1=\frac{2+3}{2}=2.5\), and the upper-half median is \(Q_3=\frac{6+7}{2}=6.5\). 4. The values \(2.5\), \(4.5\), and \(6.5\) are not observations. They are averages of adjacent ordered observations used to locate quartile positions.

The five-number summary is \(1,2.5,4.5,6.5,8\). The quartiles and median are not observed data values; they are averages of paired middle observations.

- Check whether the required ordering permits equality. - Find each whisker length by subtracting its endpoints. - Interpret coincident summary values as overlapping parts of the display.

1. A five-number summary must be nondecreasing, not strictly increasing, and \(2\leq2\leq5\leq9\leq9\). 2. The lower whisker has length \(Q_1-\min=2-2=0\). 3. The upper whisker has length \(\max-Q_3=9-9=0\). 4. The minimum coincides with \(Q_1\), and the maximum coincides with \(Q_3\). Repeated values or concentrated tails can produce these equalities.

The summary is valid because equality is allowed. Both whiskers have length \(0\): the minimum equals \(Q_1\), and the maximum equals \(Q_3\).

- List the exact locations marked by a boxplot. - Ask what happens to individual positions between those marked values. - Different raw patterns can share the same quartiles.

1. The box shows only \(Q_1\), the median, and \(Q_3\), not the locations of individual observations within those intervals. 2. The same quartile values could arise from data spread smoothly through the box or from observations concentrated in clusters with gaps between them. 3. Therefore the boxplot cannot establish the absence of gaps or clusters inside the box.

The statement is not justified. A boxplot does not reveal within-quartile gaps or clusters.

- Use the outer endpoints for one spread measure and the box endpoints for the other. - Compare the two widths with a ratio. - Convert the resulting decimal to a percentage.

1. The range is \(55-5=50\). 2. The interquartile range is \(30-18=12\). 3. The ratio is \(\frac{12}{50}=0.24\), so the interquartile range is \(24\%\) of the full range.

Range: \(50\); interquartile range: \(12\); the interquartile range is \(24\%\) of the full range.

- Check that the transformation is monotonic over the data's domain. - Apply it to each summary value. - Keep the results in increasing order.

1. The square-root function is increasing on nonnegative values, so order is preserved. 2. Taking square roots gives \(1,2,3,4,5\).

The transformed five-number summary is \(1,2,3,4,5\).

- Compare the numerical change between each pair of adjacent labels. - A boxplot encodes spread through distance along its axis. - A repair must make the visual spacing consistent with the stated scale.

1. On a linear axis, equal physical distances must represent equal numerical differences. 2. The first two tick gaps each represent \(10\) units, but the final equal-sized gap represents \(80\) units. 3. Therefore box and whisker lengths cannot be interpreted consistently as numerical spread. 4. The display can be repaired by using proportional tick spacing on a continuous linear axis or by clearly using and labeling a different valid scale.

The axis is not linear because equal tick gaps represent differences of \(10,10,\) and \(80\). Redraw it with proportional linear spacing, or use a clearly labeled valid alternative scale.

- Compute the IQR from the two quartiles. - Use the IQR to form the lower and upper fences. - Check whether equality with a fence satisfies the rule’s strict outlier condition.

1. The IQR is \(32-20=12\). 2. The fences are \(20-1.5(12)=2\) and \(32+1.5(12)=50\). 3. Outliers are values more extreme than the fences, so values exactly on the fences are not outliers.

No. Values \(2\) and \(50\) lie exactly on the fences and are not outliers by the stated rule.

- Subtract consecutive values in the five-number summary. - Compare the numerical widths of the four quartile segments. - Distinguish quartile positions in ordered data from the number of tied observations included in a closed interval.

1. The quartile-interval lengths are \(18-10=8\), \(24-18=6\), \(40-24=16\), and \(52-40=12\). 2. The interval \([18,24]\) is shortest, so the observations represented between \(Q_1\) and the median are packed into the narrowest numerical span. 3. Quartiles divide ordered positions into quarters, but tied values can occur on both sides of a quartile position. Therefore, the number of observations whose values lie in a closed quartile interval need not be exactly \(25\%\).

The interval lengths are \(8,6,16,12\). The shortest interval is \([18,24]\), suggesting the greatest concentration between \(Q_1\) and the median. Quartiles divide ordered positions, but boundary ties can make a closed interval contain more or less than exactly \(25\%\) of the observations.

- Determine whether larger original values become larger or smaller transformed values. - The transformed minimum comes from the original maximum under a decreasing rule. - List the final summary from least to greatest.

1. The transformation is decreasing, so it reverses the order of the data. 2. Transform the original maximum, third quartile, median, first quartile, and minimum in that order: \(100-70=30\), \(100-50=50\), \(100-30=70\), \(100-20=80\), and \(100-10=90\).

The transformed five-number summary is \(30,50,70,80,90\).

- Find the middle-half spread first. - Extend the appropriate multiple of that spread below and above the quartiles. - Compare outliers and whisker endpoints with the resulting cutoffs.

1. The interquartile range is \(20-12=8\). 2. The lower fence is \(12-1.5\cdot8=0\), and the upper fence is \(20+1.5\cdot8=32\). 3. The values \(-3<0\) and \(35>32\), so both are outliers. 4. The whisker endpoints \(2\) and \(30\) lie within the fences, so they are plausible most-extreme nonoutliers.

The fences are \(0\) and \(32\). Both \(-3\) and \(35\) are correctly marked as outliers.

- Compute each interquartile range in data units first. - Convert each numerical interval to its displayed length using that plot's scale. - Separate the size of a statistic from the physical size of its drawing.

1. Boxplot A has \(\text{IQR}=40-20=20\) units. 2. Boxplot B has \(\text{IQR}=24-12=12\) units. 3. A's box is drawn \(\frac{20}{5}=4\,\text{cm}\) wide, while B's box is drawn \(\frac{12}{2}=6\,\text{cm}\) wide. 4. A has the larger numerical IQR, but B appears wider because the plots use different scales.

Boxplot A has the larger IQR: \(20\) units versus \(12\) units. Boxplot B appears wider: \(6\,\text{cm}\) versus \(4\,\text{cm}\). Numerical spread must be compared from the axes, not raw page width.

- Represent the two parts of the box with lengths in the stated ratio. - Their sum is the interquartile range. - Move left and right from the median after finding both lengths.

1. Let the lower half of the box have length \(a\). Then the upper half has length \(3a\). 2. The full box length is \(a+3a=24\), so \(a=6\). 3. Therefore, \(Q_1=32-6=26\) and \(Q_3=32+18=50\).

\(Q_1=26\) and \(Q_3=50\).

- Determine whether the transformation increases or decreases as the input grows. - Match the transformed minimum with the appropriate original endpoint. - List the final summary from least to greatest.

1. The reciprocal function is decreasing for positive values, so larger original observations become smaller transformed observations. 2. Transform the original maximum, third quartile, median, first quartile, and minimum in that order. 3. The transformed five-number summary is \(\frac{1}{10},\frac{1}{5},\frac{1}{4},\frac{1}{2},1\).

The transformed five-number summary is \(\left(\frac{1}{10},\frac{1}{5},\frac{1}{4},\frac{1}{2},1\right)\). The order reverses because taking reciprocals is decreasing on positive numbers.

- Determine which ordered positions define each quartile for \(16\) observations. - Use cumulative counts to locate a position's value. - You do not need to write every repeated observation.

1. The minimum is \(1\), and the maximum is \(6\). 2. For \(16\) observations, \(Q_1\) is the average of positions \(4\) and \(5\). Both positions have value \(2\), so \(Q_1=2\). 3. The median is the average of positions \(8\) and \(9\). Both positions have value \(3\), so the median is \(3\). 4. \(Q_3\) is the average of positions \(12\) and \(13\). Those values are \(4\) and \(5\), so \(Q_3=4.5\).

The five-number summary is \(1,2,3,4.5,6\).

- Move from left to right along the boxplot. - Add each stated segment length to the previous endpoint. - Check that the resulting summary values increase.

1. The first quartile is \(4+3=7\). 2. The median is \(7+5=12\). 3. The third quartile is \(12+8=20\). 4. The maximum is \(20+4=24\).

The five-number summary is \(4,7,12,20,24\).

- Distinguish “at or below” from “strictly below.” - Consider how repeated values can occupy positions on both sides of the middle. - A boxplot does not display multiplicities at the median.

1. The median divides the ordered positions so that at least half the observations are at or below it and at least half are at or above it. 2. When many observations equal \(10\), some of the middle observations and possibly many others are tied at the median. 3. Therefore fewer than half may be strictly less than \(10\); the boxplot does not determine the exact strict-inequality count.

No. Ties at the median can make fewer than half of the observations strictly less than \(10\).

- Compare consecutive values by subtraction and by division. - Identify which comparison remains constant. - Interpret equal spacing according to the stated axis scale.

1. The numerical differences are \(10-1=9\), \(100-10=90\), \(1000-100=900\), and \(10{,}000-1000=9000\), so they are not equal. 2. Each right endpoint is \(10\) times its left endpoint. 3. Equal lengths on a base-10 logarithmic axis represent equal ratios, not equal additive differences.

The numerical differences are \(9,90,900,9000\), so they are not equal. Each segment represents multiplication by \(10\).

- Repeated values can make two quartiles equal. - Follow the fence rule even when the interquartile range is zero. - Distinguish the collapsed nonoutlier display from separately plotted values.

1. The lower half is six \(5\)s, so \(Q_1=5\). 2. The upper half is \(5,5,5,5,8,9\), whose middle two values are both \(5\), so \(Q_3=5\). 3. The interquartile range is \(0\), and both fences equal \(5\). 4. Every value different from \(5\) lies beyond a fence, so \(8\) and \(9\) are outliers. 5. The box and both whiskers collapse at \(5\), with separate outlier points at \(8\) and \(9\).

\(Q_1=5\), \(Q_3=5\), \(\text{IQR}=0\), and both fences are \(5\). The modified boxplot collapses at \(5\) and marks \(8\) and \(9\) as outliers.

- Calculate the lower cutoff from the quartiles. - Compare each low value with that cutoff. - A modified whisker reaches the most extreme value that is not an outlier.

1. The interquartile range is \(20-12=8\). 2. The lower fence is \(12-1.5\cdot8=0\). 3. Since \(-3<0\), it is plotted as a separate outlier. 4. Since \(8\) is the smallest observation within the fence, the lower whisker ends at \(8\).

Plot \(-3\) as an outlier and end the lower whisker at \(8\).

- Compute the original fences before transforming anything. - Track how quartiles and the interquartile range scale. - Compare a transformed observation with correspondingly transformed cutoffs.

1. The original interquartile range is \(9-5=4\), so the fences are \(5-1.5\cdot4=-1\) and \(9+1.5\cdot4=15\). 2. The transformed quartiles are \(15\) and \(27\), with interquartile range \(12\). 3. The transformed fences are \(15-1.5\cdot12=-3\) and \(27+1.5\cdot12=45\). 4. The fences and every observation are multiplied by the same positive factor, so outlier status is preserved.

Original fences: \(-1\) and \(15\). Transformed fences: \(-3\) and \(45\). Outlier status is preserved.

- Compare the middle-half width with the full data width. - Quartiles must lie between the minimum and maximum. - Ask when a contained interval can have the same length as the interval containing it.

1. The full range is \(20-6=14\), equal to the interquartile range. 2. Because \(Q_1\geq6\) and \(Q_3\leq20\), the difference \(Q_3-Q_1\) cannot equal the full range unless \(Q_1=6\) and \(Q_3=20\). 3. The lower whisker length is \(6-6=0\), and the upper whisker length is \(20-20=0\).

\(Q_1=6\), \(Q_3=20\), and both whisker lengths are \(0\). Equality of IQR and range forces the box to span the entire data range.

- Compute the outlier fences from the box endpoints. - Compare both extremes with those fences. - Modified and standard whiskers differ only when outliers are present.

1. The interquartile range is \(30-20=10\). 2. The fences are \(20-1.5\cdot10=5\) and \(30+1.5\cdot10=45\). 3. Both the minimum \(8\) and maximum \(42\) lie within the fences, so there are no outliers. 4. Both types of boxplot use whisker endpoints \(8\) and \(42\).

Yes. Both boxplots have whiskers at \(8\) and \(42\) because there are no outliers.

- Find the quartiles and fences before deciding where the whiskers end. - After ordering the data, apply the specified median-of-halves convention consistently. - Use the fences to identify outliers; place whiskers at the most extreme remaining data values.

1. \(Q_1=7\), median \(=9.5\), and \(Q_3=12\). 2. The IQR is \(5\), giving fences \(-0.5\) and \(19.5\). 3. The nonoutlier endpoints are \(4\) and \(13\). 4. The value \(30\) is an outlier.

\(Q_1=7\), median \(=9.5\), \(Q_3=12\); whiskers at \(4\) and \(13\); outlier \(30\).

- Complete the primary five-number analysis before changing the data set. - Treat the second calculation as a comparison, not as automatic deletion of a valid observation. - Recompute ordered positions from the beginning after the temporary removal.

1. For the nine values, the median is \(5\), \(Q_1=\frac{2+3}{2}=2.5\), and \(Q_3=\frac{7+8}{2}=7.5\). 2. The interquartile range is \(5\), so the fences are \(2.5-1.5\cdot5=-5\) and \(7.5+1.5\cdot5=15\). Thus \(30\) is an outlier. 3. After temporarily removing \(30\), the eight values have median \(4.5\), \(Q_1=2.5\), and \(Q_3=6.5\). 4. The new interquartile range is \(4\), so the new fences are \(2.5-1.5\cdot4=-3.5\) and \(6.5+1.5\cdot4=12.5\). 5. The verified value \(30\) should remain in the primary analysis. The sensitivity analysis differs because removing an observation changes ordered positions, quartiles, the interquartile range, and therefore the fences.

a) \(Q_1=2.5\), median \(=5\), \(Q_3=7.5\); fences: \(-5\) and \(15\); outlier: \(30\). b) In the sensitivity analysis, \(Q_1=2.5\), median \(=4.5\), \(Q_3=6.5\); fences: \(-3.5\) and \(12.5\). The verified value \(30\) remains part of the primary data set.

- Compute the five summary positions separately from the mean. - Internal values can move without changing a quartile boundary. - Compare what information a boxplot keeps with what it discards.

1. For both sets, the minimum is \(0\), the median is \(4\), and the maximum is \(8\). 2. In each set, the median of the lower three values is \(2\), and the median of the upper three values is \(6\). Thus both summaries are \(0,2,4,6,8\). 3. The mean of A is \(\frac{28}{7}=4\). 4. The mean of B is \(\frac{29.8}{7}\approx4.257\). 5. Identical boxplots can hide different within-quartile arrangements and different means.

Both five-number summaries are \(0,2,4,6,8\). The means are \(4\) for A and approximately \(4.257\) for B, so the same boxplot does not determine the raw distribution or mean.

- Use the quartiles to locate the high-outlier cutoff. - Interpret the whisker as the most extreme observation still inside the fences. - Separate the nonoutlier interval from the region where outlier symbols may occur.

1. The interquartile range is \(18-10=8\). 2. The upper fence is \(18+1.5\cdot8=30\). 3. The upper whisker ends at the largest nonoutlier, \(24\). Since \(27\leq30\), a value of \(27\) would be a nonoutlier and would force the whisker to extend beyond \(24\). Therefore \(27\) cannot be present. 4. No observations can lie in \((24,30]\). Values greater than \(30\) may appear as plotted outliers.

a) The upper fence is \(30\). b) No. A value of \(27\) would be a nonoutlier larger than the displayed whisker endpoint. c) The interval \((24,30]\) contains no observations.

- Compute each summary from its own ordered observations first. - Pooling requires reordering all observations together. - Compare the actual pooled positions with a componentwise numerical average.

1. Group A has summary \(0,0,0,50,100\). Group B has summary \(100,100,100,100,100\). 2. The pooled ordered data contain four \(0\)s followed by six \(100\)s. 3. The pooled median is \(100\), the lower-half median is \(0\), and the upper-half median is \(100\). Thus the pooled summary is \(0,0,100,100,100\). 4. Averaging corresponding entries gives \(50,50,50,75,100\), which differs from the pooled summary.

a) A: \(0,0,0,50,100\); B: \(100,100,100,100,100\) b) Pooled: \(0,0,100,100,100\) c) Componentwise averaging gives \(50,50,50,75,100\), so five-number summaries cannot be pooled by averaging their entries.

- Place the minimum, median, and maximum first. - Build four lower values whose middle pair averages to the first quartile. - Build four upper values whose middle pair averages to the third quartile.

1. Choose minimum \(0\), median \(5\), and maximum \(10\). 2. Make the median of the four lower observations equal to \(2\), for example with \(0,2,2,4\). 3. Make the median of the four upper observations equal to \(8\), for example with \(6,8,8,10\). 4. One valid ordered data set is \(0,2,2,4,5,6,8,8,10\).

One possible data set is \(0,2,2,4,5,6,8,8,10\).