Quelltext der Seite English:Statistics charts averages and spread - Mathematics Year 7

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= Introduction =

[[File:Graph of monthly temperature variations in south-west England.png|600px|center|alt=A month-by-month temperature graph with several red average-maximum and blue average-minimum lines.|The source distinguishes maxima/minima by its legend,but does not label each individual line with a separate location or reference period. It supports chart-reading questions,not invented precise station comparisons or current-weather claims.]]

Can two groups have the same mean yet very different distributions? Calculate centre and spread, choose a suitable chart and ask what the data actually support.

You will calculate mean, median, mode and range for three original fictional data sets, build a frequency table and justify a chart choice. Start with 3, 4, 4, 7, 12: which summary describes the middle, and which reacts strongly to the largest value?


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= Choose charts for the data and question =

Data answer a question about defined observations. Before choosing a chart, identify the variable, its unit and how values were obtained. Categorical data such as a chosen transport type are naturally displayed in separated bar-chart categories. Numeric measurements such as waiting times allow summaries of centre and spread. Our small sets below are fictional teaching values in minutes, not records of named learners or real service performance.

'''Choose a representation for its purpose.''' A frequency table shows how often each value or category appears. A bar chart compares discrete categories; its bar heights represent frequencies on a labelled scale. A pictogram can be useful for small counts, but it needs a clear key and an unambiguous interpretation of any partial symbol. Do not mistake one icon for one observation unless the key says so.

A line graph connects values in an ordered relationship, such as change over time. The source temperature graph has months on the horizontal axis and degrees Celsius vertically. Red means average maximum and blue means average minimum according to the legend. Individual curves within each colour have no separate location labels or reference period in this image; do not invent an exact named-station comparison or treat it as today’s forecast.

A histogram groups continuous numerical data into intervals. Its adjacent rectangles represent a continuous scale, rather than unrelated categories. With unequal interval widths, frequency is proportional to rectangle area and height is frequency density, not raw frequency. Simply making bar-chart bars touch does not by itself create a correctly defined histogram. This introductory distinction helps you ask the right questions before calculating.

[[File:PieChartFractionFourthsSplit.svg|420px|center|alt=A circle divided into four equal orange sectors.|Each quarter can represent25%of one declared whole. All four sectors are orange;the image does not already identify different transport categories or encode our three numerical data sets.]]

A pie chart compares shares of a complete whole. Four equal categories would each have 25%, but unequal categories need different angles. Use categories that partition the stated whole without overlap. Select a table, bar, line or pie representation because it suits the data and question, not because the graphic looks more elaborate.


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= Calculate centre and spread for three sets =

The mean is the total divided by the number of observations. The median is the middle value after ordering, or the mean of the two middle values for an even count. A mode is a most frequent value; there can be a tie. The range is maximum minus minimum and describes one aspect of spread, not a typical central value.

'''Three original data sets in minutes.'''

{| class="wikitable"
! Set !! Values !! Mean !! Median !! Mode or modes !! Range
|-
| A || 3, 4, 4, 7, 12 || 6 || 4 || 4 || 9
|-
| B || 4, 5, 6, 7, 8 || 6 || 6 || No unique mode || 4
|-
| C || 2, 2, 5, 5, 8, 8 || 5 || 5 || 2, 5 and 8 tied || 6
|}

For A, the total is 30 and there are 5 observations, so <math>\bar{x}=\frac{3+4+4+7+12}{5}=6</math>. The ordered middle is 4; the most frequent value is 4; the range 12−3=9. The mean need not equal a listed observation or the middle item. Ordering matters for the median, but not for the sum used by the mean.

For B, the same total 30 and count 5 give mean 6; its middle is 6 and range 8−4=4. All values occur once, so there is no uniquely most frequent value; under the usual school convention we report no mode rather than selecting one arbitrarily. For C, the total 30 and count 6 give mean 5; the two middle values are 5 and 5, so median 5. Values 2, 5 and 8 each occur twice and tie as modes. Its range is 8−2=6.

'''Build and check a frequency table for A.'''

{| class="wikitable"
! Value in minutes !! Frequency !! Value times frequency
|-
| 3 || 1 || 3
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| 4 || 2 || 8
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| 7 || 1 || 7
|-
| 12 || 1 || 12
|}

The frequencies total 5 and the weighted products total 30. Therefore <math>\bar{x}=\frac{3\cdot1+4\cdot2+7\cdot1+12\cdot1}{1+2+1+1}=6</math>. Dividing the sum of the four distinct value labels by 4 would discard the repeated 4 and give 6.5, a different calculation. Frequency is part of the data, not decorative information.

{{#ev:youtube|QqXLJRMxelk}}

Use Corbettmaths Primary’s original mean calculation to check the total-and-count method. Then calculate all four different summaries for our three separate sets; the video’s four values are not the data in our table. Original: corbettmaths, The Mean - Primary. The written explanations and native tasks work without sound or the external player.

Choose the median positions from the number of observations, not the number of distinct labels. A frequency table can efficiently retain repeated values without writing every item again, but the reasoning must still represent all observations.


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= Compare distributions and evaluate chart claims =

'''Compare centre with spread.''' A and B both have mean 6 minutes, but A has median 4 and range 9, while B has median 6 and range 4. The equal mean does not imply identical distributions. Neither group is automatically “better”without a defined question: lower waiting times may be desirable in one fictional context, but fairness, precision or another aim may matter in another.

'''A changed extreme value.''' Replace A’s 12 by 32, keeping 3, 4, 4, 7 unchanged. The new total is 50, mean 10, median 4, mode 4 andrange 29. The mean and range react strongly to this particular high value, while the ordered middle stays 4. This example explains sensitivity; it does not prove that the median never changes when any value is altered.

For a quick centre description in this skewed small set, the median can be informative. The mean remains useful when the total amount or equal-share interpretation matters. State which summary you use and why; reporting only the one that makes a preferred argument look strongest can mislead. Range uses only the extremes and does not describe how all intermediate values are distributed.

'''An original categorical survey model.'''
<pre>Walk: 6    Cycle: 3    Bus: 3
Total:12fictional responses,one choice per response.
Suggested chart:labelled separated bars with counts6,3,3.
Optional pie angles:180°,90°,90°.</pre>

The mode category is Walk; mean transport type is not a meaningful numeric calculation. A frequency table or separated bar chart communicates counts directly. A pictogram with one icon=3 responses would need 2, 1, 1 icons. If a sample came only from a cycling club, it would not automatically represent a whole school’s transport choices. Our counts are invented and do not describe a real survey.

'''Inspect presentation and sampling.''' Check where a vertical scale starts, its interval sizes, labels and whether sample sizes match. A truncated scale can exaggerate a small difference in bar heights. A larger observed count can reflect a larger sample rather than a greater proportion. An online poll or convenience sample does not establish a representative population result just because a chart is attractive.

Retain the three complete summary rows, frequency-table totals, a justified chart and a corrected first error. After a genuine break, recompute a median from a new ordered set and keep your actual response. Calculation accuracy and responsible interpretation are both part of statistical reasoning.


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= Interactive Tasks =


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== Quiz: Test Your Knowledge ==

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What is the mean of 3, 4, 4, 7, 12?
(6)
(!4)
(!9)
(!7.5)
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What is the median of that set?
(4)
(!6)
(!7)
(!3)
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What is its range?
(9)
(!12)
(!6)
(!4)
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What is the median of 2, 2, 5, 5, 8, 8?
(5)
(!2)
(!8)
(!6)
{{E}}

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How should the tied most frequent values in setC be reported?
(2, 5 and 8 all tie)
(!Only 5 because it is the median)
(!Only 2 because it is first)
(!Only 8 because it is largest)
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What is the frequency total in A’s table?
(5)
(!4)
(!30)
(!26)
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What happens to A’s mean when 12 is replaced by 32?
(It becomes 10)
(!It remains 6)
(!It becomes 4)
(!It becomes 29)
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What stays unchanged in that particular altered set?
(Median 4 andmode 4)
(!Mean 6 andrange 9)
(!Maximum 12)
(!Total 30)
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Which chart naturally compares the transport categories?
(Separated labelled bars)
(!A continuous time line between category names)
(!A histogram with invented numeric intervals)
(!An unlabelled pictogram without a key)
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What do unequal-width histogram rectangle areas represent?
(Frequencies proportional to areas)
(!Raw frequency always equals height)
(!Alphabetic category order)
(!The median of each interval only)
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== Memory Game ==

<div class="memo-quiz">
{|
|-
| Mean || Total divided by observation count
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| Median || Middle after ordering the observations
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| Mode || A most frequent value or tied values
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| Range || Maximum minus minimum
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| Frequency || Number of occurrences
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| Sample || Observed subset from a defined population
|}
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== Drag and Drop ==

<div class="lueckentext-quiz">
{|
|-
| '''Mean 6''' || Centre from A’s total 30 divided by 5
|-
| '''Median 4''' || Ordered middle value in A
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| '''Mode 4''' || Most frequent value in A
|-
| '''Range 9''' || A’s maximum 12 minusminimum 3
|-
| '''Frequency 2''' || Occurrence count of value 4 in A
|}
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== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| MEAN || Total divided by count
|-
| MEDIAN || Ordered middle
|-
| MODE || Most frequent value
|-
| RANGE || Maximum minus minimum
|-
| SAMPLE || Subset used to gather observations
|-
| SCALE || Labelled numerical intervals on an axis
|}
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== LearningApps ==

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.
<iframe> https://learningapps.org/index.php?s=averages+and+spread </iframe>

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== Cloze Text ==

<quiz display=simple>
{'''Complete the explanation.'''<br>
|type="{}"}
The mean divides the total by the observation { count }. The median needs data in { order }. A mode is most { frequent }. Range is maximum minus { minimum }. SetAhas mean { 6 } and median { 4 }. A chart needs a labelled { scale }. A convenience sample is not automatically { representative }.
</quiz>

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= Open Tasks =


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=== Easy ===

# [[English:Sort and locate the middle|Sort and locate the middle]]: Identify A’s median after checking the order.
# [[English:Separate centre and spread|Separate centre and spread]]: Explain why range is not another name for mean.
# [[English:Count repeated observations|Count repeated observations]]: Complete the frequencies for 3, 4, 7, 12 in A.
# [[English:Read the real graph’s limits|Read the real graph’s limits]]: Describe axes and legend without inventing names for its individual curves.

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=== Standard ===

# [[English:Calculate twelve summaries|Calculate twelve summaries]]: Give mean, median, mode and range for all three original sets.
# [[English:Check the weighted mean|Check the weighted mean]]: Use frequency totals 5 andweightedtotal 30 and explain why distinct-value averaging is wrong.
# [[English:Compare equal means|Compare equal means]]: Compare A and B using centre and spread with a stated purpose.
# [[English:Choose and construct a chart|Choose and construct a chart]]: Use the fictional transport counts and justify table, bar, pictogram or pie choices.

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=== Advanced ===

# [[English:Analyse a changed extreme|Analyse a changed extreme]]: Replace 12 by 32 and recalculate all four measures with interpretation.
# [[English:Evaluate a sampling claim|Evaluate a sampling claim]]: Explain why cycling-club responses cannot automatically represent a whole school.
# [[English:Distinguish chart types|Distinguish chart types]]: Explain categorical bars, time lines and continuous interval histograms, including unequal-width area meaning.
# [[English:Correct and retrieve later|Correct and retrieve later]]: Correct mean/median and count errors, then retain an actual later summary calculation.

{{:Open Task - Create a MOOC}}

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= Learning Check =

# [[English:Three complete summary sets|Three complete summary sets]]: Calculate all twelve original summaries.
# [[English:Frequency table|Frequency table]]: Check the original A table and its mean.
# [[English:Compare and alter a distribution|Compare and alter a distribution]]: Compare A/B and replace A’s 12 by 32.
# [[English:Chart selection|Chart selection]]: Represent the fictional transport responses.
# [[English:Graph and sample critique|Graph and sample critique]]: Inspect axes, intervals and population claims.
# [[English:Reasoning and later evidence|Reasoning and later evidence]]: Retain calculations, chart and actual later correction.

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== Worked checks and assessment guidance ==

# A: mean 6, median 4, mode 4, range 9. B: mean 6, median 6, no unique mode, range 4. C: mean 5, median 5, modes 2/5/8 tied, range 6. Totals 30 for each; counts 5, 5, 6.
# Values 3/4/7/12 havefrequencies 1/2/1/1, total 5. Products 3/8/7/12 total 30; weightedmean 6. Averaging distinct labels gives 6.5 andloses repeated observations.
# A/Bsame mean 6 butdifferent medians 4/6 andranges 9/4. ChangedAmean 10, median 4, mode 4, range 29. The example illustrates sensitivity, not that all possible changes leave a median fixed.
# Counts 6/3/3 total 12; separatedbarsorfrequencytablefitcategories. Pieangles 180/90/90 andpictogramicons 2/1/1 withkey 3 periconareconsistent. ModeWalk; numericmean transport type is not meaningful.
# Real source shows months/Celsiusandmaximum/minimumlegend, but not individual curve locations or reference period. Histograms represent continuous intervals; unequal-width area reflects frequency. Convenience samples and truncated scales can mislead.
# State variable, unit, count, sampling limits and why the summary/chart suits the question. Keep genuine later work; fictional examples are not real learner records or representative survey evidence.

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= Learning evidence =

Keep twelve summaries, a checked frequency table, a justified categorical chart and an actual later retry. Separate fictional data from real source-chart limits.

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= OER and Wikipedia =

[https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study Department for Education: England mathematics programme of study, Key Stage 3], geometry, measures and statistics; fluency, reasoning and problem solving. Year 7 allocation is editorial within KS3. All numerical tasks, worked answers, coordinate models and example data are original. Actual Commons files retain the creator/licence credits below; original-publisher YouTube embeds supplement the complete written lesson. Reviewed 12 November 2026.

[https://commons.wikimedia.org/wiki/File:Graph_of_monthly_temperature_variations_in_south-west_England.png Graph of monthly temperature variations in south-west England.png], Jolly Janner, Public domain. Unchanged original; caption and learning activities are original.

[https://commons.wikimedia.org/wiki/File:PieChartFractionFourthsSplit.svg PieChartFractionFourthsSplit.svg], Ezra Katz, [http://creativecommons.org/licenses/by-sa/3.0/ CC BY-SA 3.0]. Unchanged original; caption and learning activities are original.

<iframe> https://en.wikipedia.org/wiki/Descriptive_statistics </iframe>

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= Linked learning areas =

{| align=center
{{:D-Tab}}
'''[[English:Mathematics|Mathematics]]'''
# [[English:Transformations symmetry and geometric reasoning - Mathematics Year 7|Transformations symmetry and geometric reasoning - Mathematics Year 7]]
# [[English:Perimeter area volume and unit conversions - Mathematics Year 7|Perimeter area volume and unit conversions - Mathematics Year 7]]
|}

[[Category:Mathematics]]
[[Category:Year 7]]
[[Category:Key Stage 3]]
[[Category:Statistics]]
[[Category:Data]]
[[Category:Sprache:Englisch]]
[[Category:Locale:en-US]]

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