Cumulative Frequency
边玩边学
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课程笔记
Cumulative Frequency Basics
- Cumulative frequency is a running total of frequencies as you go through the groups.
- In a grouped frequency table, add frequencies sequentially to get cumulative frequencies.
- Cumulative frequency can be shown in an extra column or in a separate table with upper bounds.
- To find individual frequency from cumulative, subtract previous cumulative from current.
- The final cumulative frequency equals the total number of data values.
Drawing Cumulative Frequency Diagrams
- Plot cumulative frequency against the upper bound of each class interval.
- Always include a starting point at the lower bound of the first interval with cumulative frequency 0.
- Join the points with a smooth curve (not straight line segments).
- The curve typically has a stretched 'S' shape and never decreases.
- Label axes: horizontal for the variable, vertical for cumulative frequency.
Cumulative frequency diagram for quiz times
Finding Median from Cumulative Frequency Diagram
- Median position = \frac{n}{2} where n is total data values.
- Draw a horizontal line from \frac{n}{2} on the cumulative frequency axis to the curve.
- From the intersection, draw a vertical line down to the horizontal axis.
- Read the value on the horizontal axis as the median estimate.
Reading the median from the curve
Finding Quartiles and Interquartile Range
- Lower quartile (LQ) position = \frac{n}{4} .
- Upper quartile (UQ) position = \frac{3n}{4} .
- Use the same horizontal-to-vertical method as for median to find LQ and UQ.
- Interquartile range (IQR) = UQ − LQ.
- IQR measures the spread of the middle 50% of data.
Reading the quartiles from the curve
Finding Percentiles
- The p th percentile position = \frac{np}{100} .
- For example, 60th percentile position = \frac{60n}{100} .
- Draw horizontal line from that position to the curve, then vertical to axis.
- The 25th and 75th percentiles are the same as LQ and UQ; 50th is median.
Reading a percentile from the curve
Estimating Frequencies Above or Below a Value
- To find number of data values less than a given value: draw vertical line up from that value to curve, then horizontal to cumulative frequency axis, read off.
- To find number greater than a value: subtract the cumulative frequency from total n .
- For percentages, divide the count by n and multiply by 100.
Reading a frequency from the curve
Interpreting Cumulative Frequency Diagrams
- Cumulative frequency diagrams are used with grouped continuous data.
- All estimates (median, quartiles, percentiles) are approximations because raw data is unknown.
- The curve assumes data is smoothly distributed within each interval.
- Use the diagram to compare distributions (e.g., medians and IQRs).
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练习题
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1.In a cumulative frequency table, what does the cumulative frequency represent?
Easy- AThe frequency of a single class interval
- BThe running total of frequencies up to the end of each class interval
- CThe total number of data values in the whole data set
- DThe average of the frequencies
2.When drawing a cumulative frequency diagram, the cumulative frequency is plotted against which value?
Easy- AThe lower bound of the class interval
- BThe midpoint of the class interval
- CThe upper bound of the class interval
- DThe class width
3.For 80 data values, what is the position of the median on a cumulative frequency diagram?
Easy- A40
- B80
- C20
- D60
4.The cumulative frequency diagram for 100 items shows that at 50 seconds the cumulative frequency is 70. How many items took more than 50 seconds?
Medium- A70
- B30
- C50
- D20
5.From a cumulative frequency diagram of 80 items, the lower quartile is found at 12 minutes and the upper quartile at 28 minutes. What is the interquartile range?
Medium- A16 minutes
- B40 minutes
- C8 minutes
- D20 minutes
6.A cumulative frequency table has entries: 0 < t ≤ 10: CF=5, 0 < t ≤ 20: CF=18, 0 < t ≤ 30: CF=40. What is the frequency of the interval 10 < t ≤ 20?
Medium- A13
- B18
- C5
- D23
7.For 120 data values, what is the position of the 60th percentile?
Easy- A60
- B72
- C120
- D6
8.The cumulative frequency diagram for 90 bags of sweets shows that the upper quartile is 55 grams. Approximately how many bags weigh more than 55 grams?
Medium- A22.5
- B45
- C67.5
- D90