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class."},{"text":"The lower bound.","correct":false,"explanation":"The lower bound is used only for the initial zero-frequency point. Each subsequent cumulative frequency is plotted against the corresponding class’s upper bound."},{"text":"The class midpoint.","correct":false,"explanation":"Class midpoints are commonly used for frequency polygons, not cumulative frequency graphs. A cumulative frequency point is plotted at each class interval’s upper bound."},{"text":"The class width.","correct":false,"explanation":"Class width measures the size of an interval and is not a horizontal coordinate here. Cumulative frequencies are plotted against the class intervals’ upper bounds."}]},{"id":"0925be66-4ae4-4b17-8aec-33cb43becf7b","cardType":"cloze","points":null,"front":"Cumulative frequency represents a [ ] of the frequencies as you move down a table.","back":"Cumulative frequency represents a running total of the frequencies as you move down a table.","mcqOptions":[{"text":"running total","correct":true,"explanation":"Each entry adds the current class frequency to all preceding frequencies, producing a running total that increases or remains unchanged down the table."},{"text":"running average","correct":false,"explanation":"A running average would involve dividing totals by the number of values or groups considered. Cumulative frequency simply forms a running total of frequencies."},{"text":"class frequency","correct":false,"explanation":"A class frequency counts observations in one interval only. Cumulative frequency adds that count to all earlier class frequencies, forming a running total."},{"text":"frequency density","correct":false,"explanation":"Frequency density is frequency divided by class width and is used for histograms. Cumulative frequency instead represents a running total of the frequencies."}]},{"id":"d1e14794-0848-4b5b-85ef-191de3fe88f2","cardType":"basic","points":1,"front":"How can you verify that the last entry in a cumulative frequency table is correct?","back":"It must equal the total frequency ($n$) of the data set.","mcqOptions":[{"text":"It must equal the total frequency ($n$) of the data set.","correct":true,"explanation":"The final cumulative entry includes every class frequency, so it must count every observation and therefore equal the data set’s total frequency, $n$."},{"text":"It must equal the highest class frequency ($n$) of the data set.","correct":false,"explanation":"The largest individual class frequency includes only one interval and need not include every observation. The final cumulative entry must equal the total frequency, $n$."},{"text":"It must equal the number of class intervals ($n$) of the data set.","correct":false,"explanation":"Counting intervals gives the number of groups, not the number of observations. The final cumulative frequency must equal the total frequency, $n$, of the data set."},{"text":"It must equal the largest data value ($n$) of the data set.","correct":false,"explanation":"The largest data value measures magnitude rather than how many observations there are. The final cumulative entry must equal the total frequency, $n$."}]},{"id":"b862e335-e3f9-40b4-83e4-a68cccac5615","cardType":"basic","points":2,"front":"Why must a cumulative frequency curve never slope downwards?","back":"Cumulative frequency is a running total; since frequencies cannot be negative, the total only increases or stays flat.","mcqOptions":[{"text":"Cumulative frequency is a running total; since frequencies cannot be negative, the total only increases or stays flat.","correct":true,"explanation":"Adding each non-negative class frequency to the previous total can only increase it or leave it unchanged, so a cumulative frequency curve cannot slope downwards."},{"text":"Cumulative frequency is a moving average; since averages cannot be negative, the graph only increases or stays flat.","correct":false,"explanation":"Averages can rise or fall even when all values are non-negative. The graph cannot slope down because cumulative frequency is a running total of non-negative frequencies."},{"text":"Cumulative frequency records each class frequency; since every class has more values than the previous one, the graph only increases or stays flat.","correct":false,"explanation":"Successive classes do not necessarily contain increasing numbers of values. The curve is non-decreasing because it displays a running total, not separate class frequencies."},{"text":"Cumulative frequency is a running total; if a class has zero frequency, the total must decrease before increasing.","correct":false,"explanation":"A zero class frequency adds nothing, so the cumulative total stays flat rather than decreasing. Non-negative frequencies mean the curve can only rise or remain level."}]},{"id":"66921e44-656e-4aeb-b18f-c2777a3358cb","cardType":"cloze","points":null,"front":"When starting a cumulative frequency graph, you should plot a point at the [ ] with a cumulative frequency of 0.","back":"When starting a cumulative frequency graph, you should plot a point at the lowest boundary with a cumulative frequency of 0.","mcqOptions":[{"text":"lowest boundary","correct":true,"explanation":"Before the first interval begins, no observations have been counted. Plotting zero at the lowest boundary anchors the cumulative frequency curve correctly."},{"text":"upper boundary","correct":false,"explanation":"The first upper boundary is paired with the cumulative frequency of the first class, not zero. The zero-frequency starting point belongs at the lowest boundary."},{"text":"class midpoint","correct":false,"explanation":"A midpoint lies inside a class and is used in representations such as frequency polygons. The cumulative frequency curve starts at the lowest boundary."},{"text":"highest boundary","correct":false,"explanation":"The highest boundary is paired with the final cumulative frequency, which equals the total number of observations. The graph starts at zero at the lowest boundary."}]},{"id":"8be40053-27d8-463d-a7d7-9a370e13c034","cardType":"cloze","points":null,"front":"When starting a cumulative frequency graph, you should plot a point at the lowest boundary with a cumulative frequency of [ ].","back":"When starting a cumulative frequency graph, you should plot a point at the lowest boundary with a cumulative frequency of 0.","mcqOptions":[{"text":"0","correct":true,"explanation":"At the lowest boundary, none of the observations has yet been included, so the cumulative frequency is zero and the curve begins at this point."},{"text":"1","correct":false,"explanation":"This off-by-one error assumes counting begins with the first observation. Below the lowest boundary, no observations have been counted, so the cumulative frequency is 0."},{"text":"-1","correct":false,"explanation":"Frequencies count observations and cannot be negative. Before the first class interval, the running total contains no observations, so the starting cumulative frequency is 0."},{"text":"$$n$","correct":false,"explanation":"The total frequency $n$ belongs at the highest class boundary after all observations have been counted. At the lowest boundary, the cumulative frequency is 0."}]},{"id":"afc8bdce-ff79-445e-ab09-b10f3e307585","cardType":"basic","points":1,"front":"At which cumulative frequency do you find the median if the total frequency is $n$?","back":"$$\\frac{n}{2}$","mcqOptions":[{"text":"$$\\frac{n}{2}$","correct":true,"explanation":"The median divides the ordered data into two equal halves, so it is located by reading from cumulative frequency $\\frac{n}{2}$ across to the curve."},{"text":"$$\\frac{n}{4}$","correct":false,"explanation":"This position identifies the lower quartile, not the median. The median is halfway through the ordered data and is found at cumulative frequency $\\frac{n}{2}$."},{"text":"$$\\frac{3n}{4}$","correct":false,"explanation":"This position identifies the upper quartile, not the median. The median divides the data into two equal halves and is found at cumulative frequency $\\frac{n}{2}$."},{"text":"$$n$","correct":false,"explanation":"This is the final cumulative frequency and corresponds to the end of the data set. The median is halfway through, at cumulative frequency $\\frac{n}{2}$."}]},{"id":"39d649df-f525-4d4b-8f5e-c74ad482124e","cardType":"basic","points":2,"front":"For a total frequency $n$, at what cumulative frequency positions are the lower and upper quartiles found?","back":"

Lower Quartile ($Q_1$): $\\frac{n}{4}$
Upper Quartile ($Q_3$): $\\frac{3n}{4}$

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

Lower Quartile ($Q_1$): $\\frac{n}{4}$
Upper Quartile ($Q_3$): $\\frac{3n}{4}$

","correct":true,"explanation":"Quartiles divide ordered data into four equal parts. The lower quartile is one-quarter through the data, while the upper quartile is three-quarters through."},{"text":"

Lower Quartile ($Q_1$): $\\frac{n}{3}$
Upper Quartile ($Q_3$): $\\frac{2n}{3}$

","correct":false,"explanation":"These positions divide the data into thirds rather than quarters. Quartiles are found at cumulative frequencies $\\frac{n}{4}$ and $\\frac{3n}{4}$."},{"text":"

Lower Quartile ($Q_1$): $\\frac{3n}{4}$
Upper Quartile ($Q_3$): $\\frac{n}{4}$

","correct":false,"explanation":"This reverses the two quartile positions. The lower quartile occurs one-quarter through the ordered data, and the upper quartile occurs three-quarters through."},{"text":"

Lower Quartile ($Q_1$): $\\frac{n}{2}$
Upper Quartile ($Q_3$): $n$

","correct":false,"explanation":"These are the median and the final cumulative frequency, respectively. The lower and upper quartiles are located at $\\frac{n}{4}$ and $\\frac{3n}{4}$."}]},{"id":"c0620b7b-80c2-466b-9732-dc239e61b580","cardType":"basic","points":2,"front":"How do you estimate the number of data values greater than a specific value $x$ from a graph?","back":"Subtract the cumulative frequency at $x$ from the total frequency.","mcqOptions":[{"text":"Subtract the cumulative frequency at $x$ from the total frequency.","correct":true,"explanation":"The cumulative frequency at $x$ counts values up to $x$. Subtracting it from the total frequency therefore estimates how many values are greater than $x$."},{"text":"Add the cumulative frequency at $x$ to the total frequency.","correct":false,"explanation":"Adding the two frequencies double-counts values up to $x$. The required estimate is found by subtracting the cumulative frequency at $x$ from the total frequency."},{"text":"Subtract the total frequency from the cumulative frequency at $x$.","correct":false,"explanation":"Reversing the subtraction would usually produce a negative result because cumulative frequency cannot exceed total frequency. Subtract the cumulative frequency at $x$ from the total."},{"text":"Divide the cumulative frequency at $x$ by the total frequency.","correct":false,"explanation":"Dividing gives a proportion rather than the number of values above $x$. Subtract the cumulative frequency at $x$ from the total frequency to obtain the estimate."}]},{"id":"c18eaf76-f761-4940-b946-b7cabb87d9eb","cardType":"cloze","points":null,"front":"The formula for the interquartile range (IQR) is $IQR = \\text{[...]}$.","back":"The formula for the interquartile range (IQR) is $IQR = \\boxed{Q_3 - Q_1}$.","mcqOptions":[{"text":"$$Q_3 - Q_1$","correct":true,"explanation":"The interquartile range is the distance between the upper and lower quartiles, so it is calculated by subtracting $Q_1$ from $Q_3$."},{"text":"$$Q_3 + Q_1$","correct":false,"explanation":"Adding the quartiles does not measure the width of the middle half of the data. The interquartile range is the difference $Q_3 - Q_1$."},{"text":"$$Q_1 - Q_3$","correct":false,"explanation":"This reverses the subtraction and would make the IQR negative whenever $Q_3>Q_1$. The correct order is upper quartile minus lower quartile: $Q_3-Q_1$."},{"text":"$$Q_3 \\div Q_1$","correct":false,"explanation":"Dividing the quartiles calculates a ratio, not the spread between them. The interquartile range is found using the difference $Q_3 - Q_1$."}]},{"id":"ccbeb7de-6a0c-450f-9e78-261c48ae2e04","cardType":"basic","points":2,"front":"Why is the interquartile range (IQR) often used instead of the range to describe spread?","back":"It measures the middle 50% of data and is less affected by extreme values (outliers).","mcqOptions":[{"text":"It measures the middle 50% of data and is less affected by extreme values (outliers).","correct":true,"explanation":"The IQR ignores the lowest and highest quarters of the ordered data. It therefore describes the central half and is more resistant to unusually extreme observations."},{"text":"It measures the entire 100% of data and is more affected by extreme values (outliers).","correct":false,"explanation":"The range, not the IQR, uses all data values and is strongly affected by outliers. The IQR measures the middle 50%, making it less sensitive to extremes."},{"text":"It measures the outer 50% of data and is more affected by extreme values (outliers).","correct":false,"explanation":"The IQR does not focus on the outer quarters, which are precisely where extreme values occur. It measures the middle 50% and is less affected by outliers."},{"text":"It measures the middle 50% of data and is unaffected by all data values (observations).","correct":false,"explanation":"The IQR is resistant to outliers rather than being unaffected by every data value. Changes within the middle 50% can alter $Q_1$, $Q_3$, and hence the IQR."}]},{"id":"12c0d098-e640-4f55-aadc-96cf30348fca","cardType":"basic","points":3,"front":"Which five values are required to construct a box plot?","back":"

Minimum
Lower Quartile ($Q_1$)
Median
Upper Quartile ($Q_3$)
Maximum

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

Minimum
Lower Quartile ($Q_1$)
Median
Upper Quartile ($Q_3$)
Maximum

","correct":true,"explanation":"A box plot displays the five-number summary: the minimum, lower quartile, median, upper quartile and maximum. These determine the whiskers, box and median line."},{"text":"

Minimum
Lower Quartile ($Q_1$)
Mean
Upper Quartile ($Q_3$)
Maximum

","correct":false,"explanation":"The mean is not part of the standard five-number summary, so it cannot replace the median. A box plot requires the minimum, both quartiles, median and maximum."},{"text":"

Minimum
Lower Quartile ($Q_1$)
Median
Range
Maximum

","correct":false,"explanation":"The range is calculated from the minimum and maximum rather than plotted as one of the five defining values. The missing required value is the upper quartile."},{"text":"

Minimum
Mode
Median
Upper Quartile ($Q_3$)
Maximum

","correct":false,"explanation":"The mode is not required to construct a box plot and need not be unique. The lower quartile is required alongside the minimum, median, upper quartile and maximum."}]},{"id":"d5851ba9-f98f-42f7-b538-6b853e594439","cardType":"cloze","points":null,"front":"On a box plot, the middle \"box\" represents the [ ], spanning from $Q_1$ to $Q_3$.","back":"On a box plot, the middle \"box\" represents the interquartile range, spanning from $Q_1$ to $Q_3$.","mcqOptions":[{"text":"interquartile range","correct":true,"explanation":"The box extends from the lower quartile $Q_1$ to the upper quartile $Q_3$. Its width therefore represents the interquartile range, $Q_3-Q_1$."},{"text":"full data range","correct":false,"explanation":"The full range extends from the minimum to the maximum and includes both whiskers as well as the box. Only the interquartile range is represented by the box."},{"text":"lower quartile value","correct":false,"explanation":"The lower quartile is a single boundary at one end of the box, not the whole region from $Q_1$ to $Q_3$. That region represents the interquartile range."},{"text":"upper whisker range","correct":false,"explanation":"The upper whisker extends from $Q_3$ towards the maximum rather than covering the middle box. The box itself represents the interquartile range."}]},{"id":"4dcc274f-b6ba-4e4b-b9df-d08aea0c9528","cardType":"basic","points":1,"front":"When comparing two box plots, what does a higher median value indicate?","back":"The average or typical value of that data set is higher.","mcqOptions":[{"text":"The average or typical value of that data set is higher.","correct":true,"explanation":"The median is the central value when the data are ordered. A higher median therefore indicates a higher typical or average value for that distribution."},{"text":"The spread or variability of that data set is greater.","correct":false,"explanation":"A higher median concerns the centre, not the spread. The IQR or range must be compared to decide whether the values are more widely dispersed."},{"text":"The maximum or largest value of that data set is higher.","correct":false,"explanation":"The median does not determine the largest observation, so a distribution can have a higher median but a lower maximum. It indicates that the typical value is higher."},{"text":"The frequency or number of values in that data set is greater.","correct":false,"explanation":"A higher median does not reveal the sample size because frequency and central value are independent. It indicates that the distribution’s typical value is higher."}]},{"id":"6b4d89ce-6938-4bf8-bec8-e5f307c9cea3","cardType":"basic","points":1,"front":"What does a smaller interquartile range (IQR) indicate when comparing two data distributions?","back":"The middle 50% of the data is more consistent (less spread out).","mcqOptions":[{"text":"The middle 50% of the data is more consistent (less spread out).","correct":true,"explanation":"The IQR measures the spread of the central half of the ordered data. A smaller IQR means those middle observations occupy a narrower interval and are more consistent."},{"text":"The middle 50% of the data is less consistent (more spread out).","correct":false,"explanation":"This reverses the interpretation of IQR: a smaller difference between $Q_1$ and $Q_3$ means less spread, so the middle 50% is more consistent."},{"text":"The middle 50% of the data is higher overall (greater in value).","correct":false,"explanation":"IQR measures spread rather than the level of the data, so it cannot show whether central values are higher. A smaller IQR means the middle 50% is more consistent."},{"text":"The outer 50% of the data is more consistent (less extreme overall).","correct":false,"explanation":"The IQR describes only the middle half and gives no direct information about how extreme the minimum or maximum is. A smaller IQR indicates greater central consistency."}]},{"id":"c6b2fa5d-2586-45a5-bde1-5cd5d7686dab","cardType":"cloze","points":null,"front":"To compare two distributions in an exam, you must mention both a measure of [ ] (the median) and a measure of spread (the IQR).","back":"To compare two distributions in an exam, you must mention both a measure of average (the median) and a measure of spread (the IQR).","mcqOptions":[{"text":"average","correct":true,"explanation":"The median describes the centre or typical level of a distribution, so it is used as a measure of average alongside the IQR as a measure of spread."},{"text":"dispersion","correct":false,"explanation":"Spread describes variability and is measured here by the IQR, not the median. The median supplies the measure of average needed for a complete comparison."},{"text":"frequency","correct":false,"explanation":"Frequency is the number of observations and is not measured by the median. The median is a measure of average, which should be compared alongside the IQR."},{"text":"skewness","correct":false,"explanation":"The median alone does not measure asymmetry or skewness. It is being used as a measure of average, while the IQR provides the measure of spread."}]},{"id":"aae8fc82-03c8-47c2-8858-9fbaec87fc02","cardType":"cloze","points":null,"front":"To compare two distributions in an exam, you must mention both a measure of average (the median) and a measure of [ ] (the IQR).","back":"To compare two distributions in an exam, you must mention both a measure of average (the median) and a measure of spread (the IQR).","mcqOptions":[{"text":"spread","correct":true,"explanation":"The IQR describes how spread out the middle 50% of the data is, while the median describes its average or central value. Comparing both gives a fuller description of each distribution."},{"text":"frequency","correct":false,"explanation":"Frequency tells you how often values occur, not how widely the data is distributed. The required second measure is spread, represented here by the IQR."},{"text":"position","correct":false,"explanation":"Position identifies where a value lies in an ordered data set, but it is not the measure being paired with the median here. The correct term is spread."},{"text":"probability","correct":false,"explanation":"Probability describes the likelihood of an event rather than the variation within a data set. The IQR is a measure of spread, so spread is the correct term."}]},{"id":"174a0f6e-bb38-4c35-bc2b-97eefafff7a0","cardType":"basic","points":2,"front":"In the class interval $20 < t \\le 30$, does a value of 20 belong to this group?","back":"No; the inequality $t > 20$ means the value must be strictly greater than 20.","mcqOptions":[{"text":"No; the inequality $t > 20$ means the value must be strictly greater than 20.","correct":true,"explanation":"The symbol $>$ excludes 20, so only values strictly greater than 20 and up to 30 belong in this interval. A value equal to 20 is not included."},{"text":"Yes; the inequality $t > 20$ means the value may be equal to or greater than 20.","correct":false,"explanation":"This treats $t > 20$ as though it were $t \\ge 20$. The correct answer is No because the lower bound is strict, excluding a value exactly equal to 20."},{"text":"No; the inequality $t \\le 30$ means the value must be strictly less than 30.","correct":false,"explanation":"Although the answer No is correct, the explanation gives the wrong reason and misreads the upper bound. The exclusion of 20 comes from $t > 20$, meaning strictly greater than 20."},{"text":"Yes; the interval $20 < t \\le 30$ includes all values from 20 to 30 20.","correct":false,"explanation":"The lower endpoint is not included in an interval written with $t > 20$. The correct answer is No because the value must be strictly greater than 20."}]},{"id":"65047862-aa4a-4937-9fab-2c0d2db58463","cardType":"basic","points":1,"front":"To find a quartile value from a cumulative frequency graph, do you start on the x-axis or the y-axis?","back":"The y-axis (the cumulative frequency axis).","mcqOptions":[{"text":"The y-axis (the cumulative frequency axis).","correct":true,"explanation":"Cumulative frequency is read vertically on the y-axis. Start at the required quartile frequency on that axis, move horizontally to the curve, then move down to the x-axis to read the value."},{"text":"The x-axis (the data-value axis).","correct":false,"explanation":"Starting on the x-axis reverses the graph-reading process. Begin with the quartile's cumulative frequency on the y-axis, then use the curve to find the corresponding data value."},{"text":"The origin (the point where the axes meet).","correct":false,"explanation":"The origin only represents zero on both axes and does not identify a quartile frequency. Start at the required cumulative frequency on the y-axis instead."},{"text":"The midpoint (the centre of the graph).","correct":false,"explanation":"A graph's centre has no fixed statistical meaning and will not generally represent a quartile. Use the relevant cumulative frequency on the y-axis to locate the quartile."}]},{"id":"1b30f9f4-0ed8-4b7d-aa80-ee4994a190e6","cardType":"cloze","points":null,"front":"If a data set has a total frequency of 80, the upper quartile ($Q_3$) is found by reading the value at cumulative frequency [ ].","back":"If a data set has a total frequency of 80, the upper quartile ($Q_3$) is found by reading the value at cumulative frequency 60.","mcqOptions":[{"text":"60","correct":true,"explanation":"The upper quartile is the 75th percentile. Calculating $0.75 \\times 80$ gives 60, so read the data value corresponding to cumulative frequency 60."},{"text":"20","correct":false,"explanation":"This uses one quarter of the total frequency, $0.25 \\times 80$, which locates the lower quartile $Q_1$. The upper quartile requires three quarters, giving cumulative frequency 60."},{"text":"40","correct":false,"explanation":"This takes half of the total frequency, identifying the median rather than the upper quartile. For $Q_3$, use 75% of 80, which is cumulative frequency 60."},{"text":"80","correct":false,"explanation":"Using the full frequency locates the maximum or final cumulative frequency, not the upper quartile. The upper quartile is at 75% of 80, which is 60."}]},{"id":"c0d544fc-6233-4426-8d0a-f5c6d7d00a1d","cardType":"basic","points":1,"front":"Why are values read from a cumulative frequency graph considered \"estimates\"?","back":"The curve is a smooth approximation drawn between known data points.","mcqOptions":[{"text":"The curve is a smooth approximation drawn between known data points.","correct":true,"explanation":"Cumulative frequency graphs use a smooth curve to estimate values between the plotted class boundaries. Because the original grouped data is not known exactly within each class, readings are estimates."},{"text":"The curve is an exact record of every individual data value.","correct":false,"explanation":"Grouped data does not show every individual value, so the curve cannot record them exactly. It is a smooth approximation between known points, making readings estimates."},{"text":"The curve is a straight-line calculation joining the class midpoints.","correct":false,"explanation":"Cumulative frequency curves are drawn through class boundaries and are usually smooth, not based on joining class midpoints with straight lines. Therefore the readings are smooth-curve estimates."},{"text":"The curve is a bar-chart representation of the original frequencies.","correct":false,"explanation":"A cumulative frequency graph is not a bar chart of the original class frequencies. Its smooth curve approximates cumulative totals between known points, so values read from it are estimates."}]},{"id":"b83d44fa-1673-4395-a1bc-a70a75e0e5b3","cardType":"basic","points":1,"front":"If classes are written as whole numbers (e.g., 10-19, 20-29), what value should be used as the upper bound for the first group?","back":"The class boundary, which would be 19.5.","mcqOptions":[{"text":"The class boundary, which would be 19.5.","correct":true,"explanation":"Whole-number classes such as 10-19 and 20-29 have a boundary at 19.5, separating adjacent groups without a gap. This boundary is used for cumulative frequency graphs."},{"text":"The class midpoint, which would be 14.5.","correct":false,"explanation":"The midpoint is the centre of the class and is not used as its upper bound. For the class 10-19, the upper class boundary is 19.5."},{"text":"The class limit, which would be 20.","correct":false,"explanation":"Using 20 as the first upper value leaves a discontinuity between whole-number classes when treating the data as continuous. The correct boundary between 10-19 and 20-29 is 19.5."},{"text":"The class width, which would be 10.","correct":false,"explanation":"Class width measures the size of an interval; it does not give the position of its upper bound. The first group's upper boundary is 19.5, halfway between 19 and 20."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Cumulative Frequency","title":"Cumulative Frequency Flashcards","topbar":{"backHref":"/igcse/maths/edexcel/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"59a70583-a675-4d0f-aec1-11367dcf44c4","walled":"$L39"}],"$L3a","$L3b"]