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set."},{"text":"The difference between consecutive frequencies.","correct":false,"explanation":"Subtracting consecutive frequencies compares individual classes rather than accumulating them. Cumulative frequency is the running total obtained by adding frequencies successively."},{"text":"The average of all the frequencies.","correct":false,"explanation":"Calculating a mean would give an average class frequency, not an accumulated count. Cumulative frequency is the running total of the frequencies."},{"text":"The highest of the individual frequencies.","correct":false,"explanation":"Selecting the largest frequency identifies the modal class rather than the accumulated total. Cumulative frequency is the running total of all frequencies reached so far."}]},{"id":"7c4b4f58-a216-3712-73a3-9c5f3b6e5968","cardType":"cloze","points":null,"front":"The final value in a cumulative frequency table must equal the [ ].","back":"The final value in a cumulative frequency table must equal the total frequency.","mcqOptions":[{"text":"total frequency","correct":true,"explanation":"The last cumulative frequency includes every observation in every class, so it must equal the total frequency of the complete data set."},{"text":"class width","correct":false,"explanation":"Class width is the difference between a class’s boundaries and is unrelated to the final accumulated count. The final cumulative frequency equals the total frequency."},{"text":"upper boundary","correct":false,"explanation":"The upper boundary is an endpoint on the horizontal axis, not a number of observations. The final cumulative frequency must equal the total frequency."},{"text":"frequency density","correct":false,"explanation":"Frequency density is calculated by dividing frequency by class width, so it is not the final accumulated count. The last cumulative frequency equals the total frequency."}]},{"id":"6c8d3e4a-5127-377a-b204-3f7c9ad5e572","cardType":"basic","points":1,"front":"When plotting a cumulative frequency graph, which value from the class interval is used for the horizontal coordinate?","back":"The upper boundary (upper bound).","mcqOptions":[{"text":"The upper boundary (upper bound).","correct":true,"explanation":"Each cumulative total counts observations up to the end of its class interval, so it is plotted against that class’s upper boundary."},{"text":"The lower boundary (lower bound).","correct":false,"explanation":"The cumulative total for a class includes values throughout that class, not merely values below its lower boundary. Each total is plotted at the upper boundary."},{"text":"The midpoint (class centre).","correct":false,"explanation":"Midpoints are commonly used for frequency polygons, which makes this a tempting confusion. Cumulative frequencies are plotted against the upper class boundaries."},{"text":"The class width (interval size).","correct":false,"explanation":"Class width measures the size of an interval rather than locating its endpoint. A cumulative frequency point uses the class’s upper boundary as its horizontal coordinate."}]},{"id":"e5c257a2-afeb-c155-0678-26290a277838","cardType":"basic","points":1,"front":"What is the starting coordinate for a cumulative frequency graph?","back":"The lower boundary of the first class at cumulative frequency 0.","mcqOptions":[{"text":"The lower boundary of the first class at cumulative frequency 0.","correct":true,"explanation":"Before the first class begins, no observations have yet been accumulated. The graph therefore starts at the first lower boundary with cumulative frequency zero."},{"text":"The upper boundary of the first class at cumulative frequency 0.","correct":false,"explanation":"By the first upper boundary, all observations in the first class have been counted, so the cumulative frequency is not zero. Start at the first lower boundary."},{"text":"The lower boundary of the first class at cumulative frequency 1.","correct":false,"explanation":"This off-by-one error assumes counting starts with one observation. Before the first class begins, none have been counted, so the starting cumulative frequency is zero."},{"text":"The midpoint of the first class at cumulative frequency 0.","correct":false,"explanation":"The midpoint lies inside the first class, where values may already have accumulated. The graph starts at the first class’s lower boundary with cumulative frequency zero."}]},{"id":"0237ce25-eb7a-d262-c440-89be4981b9a8","cardType":"cloze","points":null,"front":"A common mistake is plotting the [ ] instead of the upper boundary.","back":"A common mistake is plotting the midpoint instead of the upper boundary.","mcqOptions":[{"text":"midpoint","correct":true,"explanation":"Midpoints are used when plotting frequency polygons, but cumulative frequency points must use upper class boundaries. Confusing the two graph types is a common error."},{"text":"lower boundary","correct":false,"explanation":"Although lower boundaries are relevant to class intervals, the commonly highlighted plotting error is using the midpoint. Cumulative totals should be plotted at upper boundaries."},{"text":"class width","correct":false,"explanation":"Class width describes the size of an interval and is not normally used as a plotted horizontal coordinate. The common error is plotting the midpoint instead of the upper boundary."},{"text":"frequency density","correct":false,"explanation":"Frequency density is a vertical-axis quantity used for histograms, not a horizontal class value. The common mistake is plotting the midpoint rather than the upper boundary."}]},{"id":"0f541cfa-5e04-46cd-8551-c6122edbd6ab","cardType":"cloze","points":null,"front":"A common mistake is plotting the midpoint instead of the [ ].","back":"A common mistake is plotting the midpoint instead of the upper boundary.","mcqOptions":[{"text":"upper boundary","correct":true,"explanation":"A cumulative total represents the number of observations up to the end of a class, so each point belongs at that class’s upper boundary."},{"text":"lower boundary","correct":false,"explanation":"Using the lower boundary places the cumulative total at the beginning of the class, before that class has been counted. It belongs at the upper boundary."},{"text":"class width","correct":false,"explanation":"Class width measures the span of an interval rather than its horizontal endpoint. Cumulative frequency points are plotted using each class’s upper boundary."},{"text":"lower limit","correct":false,"explanation":"A lower limit marks the beginning of a stated class and does not show where its cumulative total is reached. The required value is the upper boundary."}]},{"id":"23ed1181-e4c6-26ab-5be3-2c27d899ccb9","cardType":"cloze","points":null,"front":"Individual points on a cumulative frequency graph should be joined with a [ ].","back":"Individual points on a cumulative frequency graph should be joined with a smooth increasing curve.","mcqOptions":[{"text":"smooth increasing curve","correct":true,"explanation":"Cumulative frequency cannot decrease as more observations are included. A smooth increasing curve represents the growing running total and allows estimates of quartiles and percentiles."},{"text":"straight-line graph","correct":false,"explanation":"Joining every point with rigid straight sections treats the estimated distribution as changing abruptly. At GCSE, cumulative frequency points should be joined by a smooth increasing curve."},{"text":"smooth decreasing curve","correct":false,"explanation":"A decreasing curve would imply that observations are removed as the measured value increases. Cumulative totals cannot fall, so the curve must increase smoothly."},{"text":"frequency polygon","correct":false,"explanation":"A frequency polygon joins frequencies plotted at class midpoints and is a different graph type. Cumulative frequency points are joined with a smooth increasing curve."}]},{"id":"bb460f90-ad02-ba86-e1fb-e40f5ff73c23","cardType":"basic","points":1,"front":"How is the vertical position of the lower quartile ($Q_1$) calculated?","back":"$$\\frac{N}{4}$ (where $N$ is the total frequency).","mcqOptions":[{"text":"$$\\frac{N}{4}$ (where $N$ is the total frequency).","correct":true,"explanation":"The lower quartile marks the point below which one quarter of the observations lie, so its vertical position is calculated as $N \\div 4$."},{"text":"$$\\frac{N}{2}$ (where $N$ is the total frequency).","correct":false,"explanation":"Dividing the total frequency by two locates the median, not the lower quartile. The lower quartile is one quarter of the data, at $\\frac{N}{4}$."},{"text":"$$\\frac{3N}{4}$ (where $N$ is the total frequency).","correct":false,"explanation":"Using three quarters of the total locates the upper quartile, $Q_3$. The lower quartile, $Q_1$, is read at cumulative frequency $\\frac{N}{4}$."},{"text":"$$\\frac{N}{10}$ (where $N$ is the total frequency).","correct":false,"explanation":"Dividing by ten locates the first decile rather than the lower quartile. Since a quartile divides the data into four parts, use $\\frac{N}{4}$."}]},{"id":"c3bcfb54-0b08-df97-49d8-62a6fbb75b47","cardType":"basic","points":1,"front":"How is the vertical position of the median calculated?","back":"$$\\frac{N}{2}$ (where $N$ is the total frequency).","mcqOptions":[{"text":"$$\\frac{N}{2}$ (where $N$ is the total frequency).","correct":true,"explanation":"The median divides the ordered data into two equal halves, so it is located at cumulative frequency $\\frac{N}{2}$, where $N$ is the total frequency."},{"text":"$$\\frac{N}{4}$ (where $N$ is the total frequency).","correct":false,"explanation":"Using $\\frac{N}{4}$ locates the lower quartile, not the median. The median is halfway through the data, so its vertical position is $\\frac{N}{2}$."},{"text":"$$\\frac{3N}{4}$ (where $N$ is the total frequency).","correct":false,"explanation":"Using $\\frac{3N}{4}$ locates the upper quartile, not the median. The median splits the data into two equal halves, giving position $\\frac{N}{2}$."},{"text":"$$\\frac{N}{3}$ (where $N$ is the total frequency).","correct":false,"explanation":"Dividing the total frequency by three would locate a third, not the median. The median is halfway through the ordered data, so its position is $\\frac{N}{2}$."}]},{"id":"74451bef-e8ca-f6be-dccb-5ec988b31928","cardType":"basic","points":1,"front":"How is the vertical position of the upper quartile ($Q_3$) calculated?","back":"$$\\frac{3N}{4}$ (where $N$ is the total frequency).","mcqOptions":[{"text":"$$\\frac{3N}{4}$ (where $N$ is the total frequency).","correct":true,"explanation":"The upper quartile has 75% of the data at or below it. Therefore, its vertical position is $\\frac{3N}{4}$, where $N$ is the total frequency."},{"text":"$$\\frac{N}{4}$ (where $N$ is the total frequency).","correct":false,"explanation":"Using $\\frac{N}{4}$ gives the lower quartile, where 25% of the data lies at or below it. The upper quartile is at $\\frac{3N}{4}$."},{"text":"$$\\frac{N}{2}$ (where $N$ is the total frequency).","correct":false,"explanation":"Using $\\frac{N}{2}$ gives the median, which is halfway through the data. The upper quartile is three-quarters through, at $\\frac{3N}{4}$."},{"text":"$$\\frac{4N}{3}$ (where $N$ is the total frequency).","correct":false,"explanation":"Reversing the fraction produces a position greater than the total frequency, so it cannot be a quartile. The upper quartile is located at $\\frac{3N}{4}$."}]},{"id":"cf315a00-7cde-da45-2d06-8cc286f61574","cardType":"basic","points":3,"front":"What are the steps to estimate a quartile from a cumulative frequency graph?","back":"

Find the position on the vertical axis
Read across to the curve
Read down to the horizontal axis

","mcqOptions":[{"text":"

Find the position on the vertical axis
Read across to the curve
Read down to the horizontal axis

","correct":true,"explanation":"A quartile position is first found using cumulative frequency on the vertical axis. Moving across to the curve and then down gives the corresponding data value."},{"text":"

Find the position on the horizontal axis
Read up to the curve
Read across to the vertical axis

","correct":false,"explanation":"This procedure starts with a data value and finds its cumulative frequency, reversing the required process. Begin at the quartile position vertically, then read across and down."},{"text":"

Find the position on the vertical axis
Read down to the curve
Read across to the horizontal axis

","correct":false,"explanation":"Reading down from the vertical axis does not lead correctly to the curve. From the quartile position, read horizontally across to the curve and then vertically down."},{"text":"

Find the position on the horizontal axis
Read across to the curve
Read up to the vertical axis

","correct":false,"explanation":"Quartile positions are cumulative frequencies, so they must be located on the vertical axis rather than the horizontal axis. Then read across to the curve and down."}]},{"id":"18097bf3-be4a-e8e0-858f-0837a8318751","cardType":"basic","points":1,"front":"How is the interquartile range (IQR) calculated?","back":"$$Q_3 - Q_1$","mcqOptions":[{"text":"$$Q_3 - Q_1$","correct":true,"explanation":"The interquartile range is the width of the middle 50% of the data. It is calculated by subtracting the lower quartile from the upper quartile."},{"text":"$$Q_3 + Q_1$","correct":false,"explanation":"Adding the quartiles does not measure the distance between them. The interquartile range is the upper quartile minus the lower quartile, so it is $Q_3 - Q_1$."},{"text":"$$Q_1 - Q_3$","correct":false,"explanation":"Reversing the subtraction usually produces a negative value, which cannot represent a spread. Subtract the lower quartile from the upper quartile: $Q_3 - Q_1$."},{"text":"$$Q_3 - Q_2$","correct":false,"explanation":"Subtracting the median $Q_2$ from the upper quartile measures only one section of the box. The full interquartile range is $Q_3 - Q_1$."}]},{"id":"e1efb5b2-6f5b-a02d-01c3-a43e43eaad25","cardType":"cloze","points":null,"front":"The interquartile range measures the spread of the [ ] of the data.","back":"The interquartile range measures the spread of the middle 50% of the data.","mcqOptions":[{"text":"middle 50%","correct":true,"explanation":"The interval from the lower quartile to the upper quartile contains the central half of the ordered data. Therefore, the interquartile range measures the spread of the middle 50%."},{"text":"lowest 50%","correct":false,"explanation":"The lowest 50% runs from the minimum to the median, not between the quartiles. The interquartile range measures the spread of the middle 50% of the data."},{"text":"middle 75%","correct":false,"explanation":"The quartiles divide the data into four equal 25% sections, and the interval from $Q_1$ to $Q_3$ covers two sections. It therefore contains the middle 50%."},{"text":"highest 25%","correct":false,"explanation":"The highest 25% lies above the upper quartile and is not measured by the interquartile range. The IQR covers the middle 50%, from $Q_1$ to $Q_3$."}]},{"id":"d12b095f-8b11-d5e5-d354-8eb65ed23bd7","cardType":"basic","points":2,"front":"Why are values read from a cumulative frequency graph called estimates?","back":"

The original data is grouped
The curve is drawn by hand

","mcqOptions":[{"text":"

The original data is grouped
The curve is drawn by hand

","correct":true,"explanation":"Grouped data does not give every exact individual value, and a hand-drawn curve introduces graphical uncertainty. Consequently, values read from the curve can only be estimates."},{"text":"

The original data is continuous
The curve is perfectly accurate

","correct":false,"explanation":"Continuous data can be recorded exactly, and a perfectly accurate curve would not itself cause uncertainty. Estimates arise because the data is grouped and the curve is drawn by hand."},{"text":"

The original data is unordered
The curve is drawn by computer

","correct":false,"explanation":"Cumulative frequency already organises observations by value, so unordered data is not the issue. Uncertainty comes from grouped intervals and reading from a hand-drawn curve."},{"text":"

The original data is discrete
The curve is made of straight lines

","correct":false,"explanation":"Discrete data is not automatically estimated, and cumulative frequency curves are not necessarily made of straight lines. The estimates result from grouping and hand-drawing the curve."}]},{"id":"a82c7ed8-92d9-7b0a-0670-0d009955b9e5","cardType":"basic","points":3,"front":"Which five values are used to construct a box plot?","back":"

Lowest value
Lower quartile ($Q_1$)
Median
Upper quartile ($Q_3$)
Highest value

","mcqOptions":[{"text":"

Lowest value
Lower quartile ($Q_1$)
Median
Upper quartile ($Q_3$)
Highest value

","correct":true,"explanation":"A box plot uses the five-number summary: the minimum, lower quartile, median, upper quartile and maximum. These values determine the whisker ends, box edges and central line."},{"text":"

Lowest value
Lower quartile ($Q_1$)
Mean
Upper quartile ($Q_3$)
Highest value

","correct":false,"explanation":"The central line of a standard box plot represents the median, not the mean. The other four values are the minimum, lower quartile, upper quartile and maximum."},{"text":"

Lowest value
Lower quartile ($Q_1$)
Median
Interquartile range
Highest value

","correct":false,"explanation":"The interquartile range determines the box’s width but is not itself one of the five plotted positions. The upper quartile $Q_3$ must be included instead."},{"text":"

Range
Lower quartile ($Q_1$)
Median
Upper quartile ($Q_3$)
Highest value

","correct":false,"explanation":"The range is a difference, not a plotted endpoint. A box plot needs the actual lowest value alongside the lower quartile, median, upper quartile and highest value."}]},{"id":"e5799b1c-e3ad-7cf3-77b1-f00e6753802d","cardType":"cloze","points":null,"front":"In a box plot, the [ ] line does not have to be in the center of the box.","back":"In a box plot, the median line does not have to be in the center of the box.","mcqOptions":[{"text":"median","correct":true,"explanation":"The line inside the box marks the median. It is centred only when the distances from the median to the lower and upper quartiles are equal."},{"text":"mean","correct":false,"explanation":"A standard box plot does not usually display the mean as its internal line. The line marks the median, which need not be centred between the two quartiles."},{"text":"range","correct":false,"explanation":"The range is the numerical difference between the highest and lowest values, not a line inside the box. The internal line represents the median."},{"text":"frequency","correct":false,"explanation":"Frequency counts how often values occur and is not represented by the line inside a standard box plot. That line shows the median of the data."}]},{"id":"5eaf4227-a3fd-f5ec-1a67-c2d8b54dde49","cardType":"basic","points":1,"front":"When comparing two data sets, what does a higher median indicate?","back":"The typical value is higher.","mcqOptions":[{"text":"The typical value is higher.","correct":true,"explanation":"The median is the middle value when the data is ordered, so a higher median shows that the typical or central value of the data set is higher."},{"text":"The typical value is lower.","correct":false,"explanation":"This reverses the comparison between the medians. A higher median means the middle, or typical, value is higher rather than lower."},{"text":"The spread of values is greater.","correct":false,"explanation":"The median measures central tendency, not spread. A higher median indicates a higher typical value; the IQR or range must be compared to assess spread."},{"text":"The data consistency is higher.","correct":false,"explanation":"Consistency is determined by measures of spread such as the IQR, not by the median. A higher median only indicates that the typical value is higher."}]},{"id":"e42a60e3-0906-6562-1247-f467b9186cc1","cardType":"basic","points":1,"front":"When comparing two data sets, what does a smaller IQR indicate?","back":"The data is more consistent (less spread out).","mcqOptions":[{"text":"The data is more consistent (less spread out).","correct":true,"explanation":"The IQR measures the spread of the middle 50% of the data. A smaller IQR means these values are closer together, indicating greater consistency."},{"text":"The data is less consistent (more spread out).","correct":false,"explanation":"This reverses the meaning of the IQR comparison. A smaller IQR means the middle 50% is less spread out, so the data is more consistent."},{"text":"The data has a higher median (larger typical value).","correct":false,"explanation":"The IQR measures spread and does not determine the median. A smaller IQR indicates more consistent data, but the typical value could be higher or lower."},{"text":"The data is more symmetrical (evenly spread out).","correct":false,"explanation":"The size of the IQR alone does not reveal the distribution’s symmetry. It only shows that the middle 50% is less spread out and therefore more consistent."}]},{"id":"572f0b1a-f33b-0760-ceab-0677eeda93db","cardType":"basic","points":1,"front":"In the class interval $10 < t \\le 20$, what is the upper boundary?","back":"20","mcqOptions":[{"text":"20","correct":true,"explanation":"The inequality includes values up to and including 20, so 20 is the upper boundary of the class interval."},{"text":"10","correct":false,"explanation":"The value 10 is the lower boundary, even though it is excluded by the strict inequality. The upper boundary is 20."},{"text":"19","correct":false,"explanation":"Choosing 19 incorrectly treats the data as restricted to whole numbers. The stated class interval extends continuously to 20, which is its upper boundary."},{"text":"21","correct":false,"explanation":"This adds one to the included endpoint, a common whole-number interval mistake. The class ends at 20, so its upper boundary is 20."}]},{"id":"1e630be7-510c-41a0-4ff6-d81b7c7ee0c8","cardType":"basic","points":1,"front":"If the total frequency ($N$) is 80, what is the vertical position for the lower quartile?","back":"20 (since $\\frac{80}{4} = 20$).","mcqOptions":[{"text":"20 (since $\\frac{80}{4} = 20$).","correct":true,"explanation":"The lower quartile is located at one quarter of the total cumulative frequency. With 80 values, its vertical position is $\\frac{80}{4}=20$."},{"text":"40 (since $\\frac{80}{2} = 40$).","correct":false,"explanation":"Dividing the total frequency by 2 finds the median position, not the lower quartile. The lower quartile is at $\\frac{80}{4}=20$."},{"text":"60 (since $\\frac{3 \\times 80}{4} = 60$).","correct":false,"explanation":"Using three quarters of the total gives the upper quartile position. The lower quartile uses one quarter, so its vertical position is $\\frac{80}{4}=20$."},{"text":"16 (since $\\frac{80}{5} = 16$).","correct":false,"explanation":"Dividing by 5 incorrectly splits the data into fifths. Quartiles divide ordered data into four parts, so the lower quartile position is $\\frac{80}{4}=20$."}]},{"id":"2c9cbbdf-e622-9d03-3b65-5b4ac78473ad","cardType":"basic","points":1,"front":"How is the range calculated from a box plot?","back":"Highest value minus the lowest value.","mcqOptions":[{"text":"Highest value minus the lowest value.","correct":true,"explanation":"The range measures the full spread of the data, from the minimum to the maximum. It is therefore calculated by subtracting the lowest value from the highest value."},{"text":"Highest value plus the lowest value.","correct":false,"explanation":"Adding the two endpoints does not measure the distance between them. The range is found by subtracting the lowest value from the highest value."},{"text":"Upper quartile minus the lower quartile.","correct":false,"explanation":"Subtracting the lower quartile from the upper quartile calculates the IQR, which measures the middle 50%. The range uses the highest and lowest values instead."},{"text":"Lowest value minus the highest value.","correct":false,"explanation":"This reverses the subtraction and usually produces a negative result. The range is non-negative and is calculated as the highest value minus the lowest value."}]}]}],"examCountdown":null,"heading":"Cumulative Frequency","title":"Cumulative Frequency Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"52e3db03-e1d0-bab7-11c4-d01a571b2a09","walled":"$L39"}],"$L3a","$L3b"]