A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency ") in mathematics , especially in statistics . Each attempts to summarize or typify a given group of data , illustrating the magnitude and sign of the data set . Which of these measures is most illuminating depends on what is being measured, and on context and purpose.
57-473: At 316 metres (1,037 ft) above sea level , Walton Hill is the highest point in the range of hills in northern Worcestershire known as the Clent Hills . It is the highest point for 21 miles (34 km) in all directions, and as such commands an excellent panorama . Its neighbours include Clent Hill, Wychbury Hill (which is hidden from view by Clent Hill), Calcot Hill, and Romsley Hill. The summit of
114-425: A color wheel —there is no mean to the set of all colors. In these situations, you must decide which mean is most useful. You can do this by adjusting the values before averaging, or by using a specialized approach for the mean of circular quantities . The Fréchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold . Unlike many other means,
171-479: A random variable having that distribution. If the random variable is denoted by X {\displaystyle X} , then the mean is also known as the expected value of X {\displaystyle X} (denoted E ( X ) {\displaystyle E(X)} ). For a discrete probability distribution , the mean is given by ∑ x P ( x ) {\displaystyle \textstyle \sum xP(x)} , where
228-400: A truncated mean . It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values. The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing
285-409: A defined barometric pressure . Generally, the pressure used to set the altimeter is the barometric pressure that would exist at MSL in the region being flown over. This pressure is referred to as either QNH or "altimeter" and is transmitted to the pilot by radio from air traffic control (ATC) or an automatic terminal information service (ATIS). Since the terrain elevation is also referenced to MSL,
342-432: A few metres, in timeframes ranging from minutes to months: Between 1901 and 2018, the average sea level rose by 15–25 cm (6–10 in), with an increase of 2.3 mm (0.091 in) per year since the 1970s. This was faster than the sea level had ever risen over at least the past 3,000 years. The rate accelerated to 4.62 mm (0.182 in)/yr for the decade 2013–2022. Climate change due to human activities
399-417: A function f ( x ) {\displaystyle f(x)} . Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely by integration . The integration formula is written as: In this case, care must be taken to make sure that
456-438: A reference datum for mean sea level (MSL). It is also used in aviation, where some heights are recorded and reported with respect to mean sea level (contrast with flight level ), and in the atmospheric sciences , and in land surveying . An alternative is to base height measurements on a reference ellipsoid approximating the entire Earth, which is what systems such as GPS do. In aviation, the reference ellipsoid known as WGS84
513-430: A sample x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} , usually denoted by x ¯ {\displaystyle {\bar {x}}} , is the sum of the sampled values divided by the number of items in the sample. For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: The geometric mean
570-463: A sharp reduction in greenhouse gas emissions, this may increase to hundreds of millions in the latter decades of the century. Local factors like tidal range or land subsidence will greatly affect the severity of impacts. For instance, sea level rise in the United States is likely to be two to three times greater than the global average by the end of the century. Yet, of the 20 countries with
627-502: A tee-junction at the top of St. Kenelm's Pass next to High Harcourt Farm. The lane from High Harcourt Farm skirts Walton Hill finishing at another tee-junction with the A491 road at the Hollybush ( public house ). For access routes please refer to the relevant Ordnance Survey map (Explorer 1:25,000 series – Wolverhampton & Dudley). Further along the lane the hill can be accessed on foot via
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#1732852861053684-417: Is an average surface level of one or more among Earth 's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datum – a standardised geodetic datum – that is used, for example, as a chart datum in cartography and marine navigation , or, in aviation, as the standard sea level at which atmospheric pressure
741-401: Is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean): For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit , as in
798-488: Is because the sea is in constant motion, affected by the tides, wind , atmospheric pressure, local gravitational differences, temperature, salinity , and so forth. The mean sea level at a particular location may be calculated over an extended time period and used as a datum . For example, hourly measurements may be averaged over a full Metonic 19-year lunar cycle to determine the mean sea level at an official tide gauge . Still-water level or still-water sea level (SWL)
855-422: Is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions. The mean of a probability distribution is the long-run arithmetic average value of
912-427: Is due to change in either the volume of water in the world's oceans or the volume of the oceanic basins . Two major mechanisms are currently causing eustatic sea level rise. First, shrinking land ice, such as mountain glaciers and polar ice sheets, is releasing water into the oceans. Second, as ocean temperatures rise, the warmer water expands. Many factors can produce short-term changes in sea level, typically within
969-483: Is increasingly used to define heights; however, differences up to 100 metres (328 feet) exist between this ellipsoid height and local mean sea level. Another alternative is to use a geoid -based vertical datum such as NAVD88 and the global EGM96 (part of WGS84). Details vary in different countries. When referring to geographic features such as mountains, on a topographic map variations in elevation are shown by contour lines . A mountain's highest point or summit
1026-512: Is measured to calibrate altitude and, consequently, aircraft flight levels . A common and relatively straightforward mean sea-level standard is instead a long-term average of tide gauge readings at a particular reference location. Sea levels can be affected by many factors and are known to have varied greatly over geological time scales . Current sea level rise is mainly caused by human-induced climate change . When temperatures rise, mountain glaciers and polar ice sheets melt, increasing
1083-487: Is the probability density function . In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure . The mean need not exist or be finite; for some probability distributions the mean is infinite ( +∞ or −∞ ), while for others the mean is undefined . The generalized mean , also known as
1140-413: Is the arithmetic average of the values; however, for skewed distributions , the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population
1197-447: Is the level of the sea with motions such as wind waves averaged out. Then MSL implies the SWL further averaged over a period of time such that changes due to, e.g., the tides , also have zero mean. Global MSL refers to a spatial average over the entire ocean area, typically using large sets of tide gauges and/or satellite measurements. One often measures the values of MSL with respect to
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#17328528610531254-723: Is the main cause. Between 1993 and 2018, melting ice sheets and glaciers accounted for 44% of sea level rise , with another 42% resulting from thermal expansion of water . Sea level rise lags behind changes in the Earth 's temperature by many decades, and sea level rise will therefore continue to accelerate between now and 2050 in response to warming that has already happened. What happens after that depends on human greenhouse gas emissions . If there are very deep cuts in emissions, sea level rise would slow between 2050 and 2100. It could then reach by 2100 slightly over 30 cm (1 ft) from now and approximately 60 cm (2 ft) from
1311-480: Is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x 1 , x 2 , ..., x n is typically denoted using an overhead bar , x ¯ {\displaystyle {\bar {x}}} . If the numbers are from observing a sample of a larger group , the arithmetic mean is termed the sample mean ( x ¯ {\displaystyle {\bar {x}}} ) to distinguish it from
1368-527: Is typically illustrated with the AMSL height in metres, feet or both. In unusual cases where a land location is below sea level, such as Death Valley, California , the elevation AMSL is negative. It is often necessary to compare the local height of the mean sea surface with a "level" reference surface, or geodetic datum, called the geoid . In the absence of external forces, the local mean sea level would coincide with this geoid surface, being an equipotential surface of
1425-595: The Parish church in Clent, and joins one of the two bridleways emanating from Walton Pool. Another path starts in the hamlet of Clatterbach, near the Vine Inn in Clent village and this allows direct access to the north western flank of Walton Hill. The northern car park is at 260 metres (850 ft) above sea level. The other access points onto the hill start from slightly lower down the slope at 250 metres (820 ft) The access from
1482-778: The Shropshire Hills , the Bromyard Downs in Herefordshire , and 55 miles (89 km) south west to the Black Mountains . This comes from its situation in the middle of England, distant from all other hills but not too distant. The hill can be walked up on foot from most directions. The most common ascent is from a car park north east of the summit. The car park is most easily accessed from A456 road via Hagley Wood Lane (leaving it between Hagley and Hayley Green ) or Uffmoor Lane (leaving it just east of Hayley Green). Both lead to
1539-400: The arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. The arithmetic mean (or simply mean or average ) of a list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of
1596-612: The geoid of the Earth, which approximates the local mean sea level for locations in the open ocean. The geoid includes a significant depression in the Indian Ocean , whose surface dips as much as 106 m (348 ft) below the global mean sea level (excluding minor effects such as tides and currents). Precise determination of a "mean sea level" is difficult because of the many factors that affect sea level. Instantaneous sea level varies substantially on several scales of time and space. This
1653-441: The group mean (or expected value ) of the underlying distribution, denoted μ {\displaystyle \mu } or μ x {\displaystyle \mu _{x}} . Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis ; examples are given below. In mathematics, the three classical Pythagorean means are
1710-418: The height above mean sea level (AMSL). The term APSL means above present sea level, comparing sea levels in the past with the level today. Earth's radius at sea level is 6,378.137 km (3,963.191 mi) at the equator. It is 6,356.752 km (3,949.903 mi) at the poles and 6,371.001 km (3,958.756 mi) on average. This flattened spheroid , combined with local gravity anomalies , defines
1767-544: The 1690s. Satellite altimeters have been making precise measurements of sea level since the launch of TOPEX/Poseidon in 1992. A joint mission of NASA and CNES , TOPEX/Poseidon was followed by Jason-1 in 2001 and the Ocean Surface Topography Mission on the Jason-2 satellite in 2008. Height above mean sea level ( AMSL ) is the elevation (on the ground) or altitude (in the air) of an object, relative to
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1824-973: The 19th century. With high emissions it would instead accelerate further, and could rise by 1.0 m ( 3 + 1 ⁄ 3 ft) or even 1.6 m ( 5 + 1 ⁄ 3 ft) by 2100. In the long run, sea level rise would amount to 2–3 m (7–10 ft) over the next 2000 years if warming stays to its current 1.5 °C (2.7 °F) over the pre-industrial past. It would be 19–22 metres (62–72 ft) if warming peaks at 5 °C (9.0 °F). Rising seas affect every coastal and island population on Earth. This can be through flooding, higher storm surges , king tides , and tsunamis . There are many knock-on effects. They lead to loss of coastal ecosystems like mangroves . Crop yields may reduce because of increasing salt levels in irrigation water. Damage to ports disrupts sea trade. The sea level rise projected by 2050 will expose places currently inhabited by tens of millions of people to annual flooding. Without
1881-448: The Earth's gravitational field which, in itself, does not conform to a simple sphere or ellipsoid and exhibits gravity anomalies such as those measured by NASA's GRACE satellites . In reality, the geoid surface is not directly observed, even as a long-term average, due to ocean currents, air pressure variations, temperature and salinity variations, etc. The location-dependent but time-persistent separation between local mean sea level and
1938-501: The Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. It is sometimes also known as the Karcher mean (named after Hermann Karcher). In geometry, there are thousands of different definitions for the center of a triangle that can all be interpreted as the mean of a triangular set of points in the plane. This is an approximation to
1995-672: The Marégraphe in Marseilles measures continuously the sea level since 1883 and offers the longest collated data about the sea level. It is used for a part of continental Europe and the main part of Africa as the official sea level. Spain uses the reference to measure heights below or above sea level at Alicante , while the European Vertical Reference System is calibrated to the Amsterdam Peil elevation, which dates back to
2052-456: The amount of water in the oceans, while the existing seawater also expands with heat. Because most of human settlement and infrastructure was built in response to a more-normalized sea level with limited expected change, populations affected by sea level rise will need to invest in climate adaptation to mitigate the worst effects or, when populations are at extreme risk, a process of managed retreat . The term above sea level generally refers to
2109-440: The case of speed (i.e., distance per unit of time): For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of 15 {\displaystyle 15} tells us that these five different pumps working together will pump at the same rate as much as five pumps that can each empty
2166-482: The geoid is referred to as (mean) ocean surface topography . It varies globally in a typical range of ±1 m (3 ft). Several terms are used to describe the changing relationships between sea level and dry land. The melting of glaciers at the end of ice ages results in isostatic post-glacial rebound , when land rises after the weight of ice is removed. Conversely, older volcanic islands experience relative sea level rise, due to isostatic subsidence from
2223-624: The greatest exposure to sea level rise, twelve are in Asia , including Indonesia , Bangladesh and the Philippines. The resilience and adaptive capacity of ecosystems and countries also varies, which will result in more or less pronounced impacts. The greatest impact on human populations in the near term will occur in the low-lying Caribbean and Pacific islands . Sea level rise will make many of them uninhabitable later this century. Pilots can estimate height above sea level with an altimeter set to
2280-551: The height of planetary features. Local mean sea level (LMSL) is defined as the height of the sea with respect to a land benchmark, averaged over a period of time long enough that fluctuations caused by waves and tides are smoothed out, typically a year or more. One must adjust perceived changes in LMSL to account for vertical movements of the land, which can occur at rates similar to sea level changes (millimetres per year). Some land movements occur because of isostatic adjustment to
2337-576: The hill is open grassland and heath , which was commonland of the manor of Clent . This became a regulated common in 1935, as a result of action by Bromsgrove Rural District Council , and was given to the National Trust in 1959. Woodlands on the side of the hill in the Clatterbach valley were bought by Worcestershire County Council in 1957 and given by it to the National Trust in 1959. From
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2394-443: The integral converges. But the mean may be finite even if the function itself tends to infinity at some points. Angles , times of day, and other cyclical quantities require modular arithmetic to add and otherwise combine numbers. In all these situations, there will not be a unique mean. For example, the times an hour before and after midnight are equidistant to both midnight and noon. It is also possible that no mean exists. Consider
2451-576: The land; hence a change in relative MSL or ( relative sea level ) can result from a real change in sea level, or from a change in the height of the land on which the tide gauge operates, or both. In the UK, the ordnance datum (the 0 metres height on UK maps) is the mean sea level measured at Newlyn in Cornwall between 1915 and 1921. Before 1921, the vertical datum was MSL at the Victoria Dock, Liverpool . Since
2508-443: The long bricked paved driveway to Walton Hill Farm. From the same lane, about halfway to the Hollybush, is Sling Common near the entrance to Calcot Hill Farm on Calcot Hill, which gives access to the summit of Walton Hill along a footpath which follows a ridge through private farm land, leading to Walton Hill Farm, where the path enters the National Trust land on the summit of Walton Hill. A bridle path runs from Calcot Hill Farm, along
2565-411: The lowest and the highest quarter of values. assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights. In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable ) set of values. This can happen when calculating the mean value y avg {\displaystyle y_{\text{avg}}} of
2622-422: The mean and size of sample i {\displaystyle i} respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values. Sometimes, a set of numbers might contain outliers (i.e., data values which are much lower or much higher than the others). Often, outliers are erroneous data caused by artifacts . In this case, one can use
2679-481: The melting of ice sheets at the end of the last ice age . The weight of the ice sheet depresses the underlying land, and when the ice melts away the land slowly rebounds . Changes in ground-based ice volume also affect local and regional sea levels by the readjustment of the geoid and true polar wander . Atmospheric pressure , ocean currents and local ocean temperature changes can affect LMSL as well. Eustatic sea level change (global as opposed to local change)
2736-497: The northern car park has the steepest and shortest route to the summit, but there is also an easy access trail which winds around the end of the hill. The Calcot Hill route is just over a mile long (about two kilometres). The other routes are all about a mile (about a kilometre and a half) to the summit. 52°24′55″N 2°05′12″W / 52.41527°N 2.08670°W / 52.41527; -2.08670 Sea level Mean sea level ( MSL , often shortened to sea level )
2793-517: The parameter m , the following types of means are obtained: This can be generalized further as the generalized f -mean and again a suitable choice of an invertible f will give The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population: Where x i ¯ {\displaystyle {\bar {x_{i}}}} and w i {\displaystyle w_{i}} are
2850-459: The pilot can estimate height above ground by subtracting the terrain altitude from the altimeter reading. Aviation charts are divided into boxes and the maximum terrain altitude from MSL in each box is clearly indicated. Once above the transition altitude, the altimeter is set to the international standard atmosphere (ISA) pressure at MSL which is 1013.25 hPa or 29.92 inHg. Mean The arithmetic mean , also known as "arithmetic average",
2907-511: The power mean or Hölder mean, is an abstraction of the quadratic , arithmetic, geometric, and harmonic means. It is defined for a set of n positive numbers x i by x ¯ ( m ) = ( 1 n ∑ i = 1 n x i m ) 1 m {\displaystyle {\bar {x}}(m)=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{m}\right)^{\frac {1}{m}}} By choosing different values for
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#17328528610532964-530: The southern face of the Walton Hill to the hamlet of Walton Pool . This crosses a footpath from Belbroughton and the A491 across farmland on its way towards the summit. From Walton Pool, a hamlet in the parish of Clent , two bridle paths follow the lines of two ridges, and meet to form the main south western spine of Walton hill. To the north west of Walton Pool there is a footpath which starts at St. Leonards church,
3021-467: The sum is taken over all possible values of the random variable and P ( x ) {\displaystyle P(x)} is the probability mass function . For a continuous distribution , the mean is ∫ − ∞ ∞ x f ( x ) d x {\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx} , where f ( x ) {\displaystyle f(x)}
3078-648: The summit looking south west down the main spine of the hill the closer peaks of the Clee Hills , The Wrekin , Malvern and Abberley hills are usually visible. On a clear day one can see 30 miles (48 km) south to the Cotswolds , 25 miles (40 km) south west to the Malvern Hills , 35 miles (56 km) north east to Charnwood Forest , 40 miles (64 km) north to the Peak District , 20 miles (32 km) west to
3135-419: The tank in 15 {\displaystyle 15} minutes. AM, GM, and HM satisfy these inequalities: Equality holds if all the elements of the given sample are equal. In descriptive statistics , the mean may be confused with the median , mode or mid-range , as any of these may incorrectly be called an "average" (more formally, a measure of central tendency ). The mean of a set of observations
3192-507: The times of the Russian Empire , in Russia and its other former parts, now independent states, the sea level is measured from the zero level of Kronstadt Sea-Gauge. In Hong Kong, "mPD" is a surveying term meaning "metres above Principal Datum" and refers to height of 0.146 m (5.7 in) above chart datum and 1.304 m (4 ft 3.3 in) below the average sea level. In France,
3249-411: The weight of cooling volcanos. The subsidence of land due to the withdrawal of groundwater is another isostatic cause of relative sea level rise. On planets that lack a liquid ocean, planetologists can calculate a "mean altitude" by averaging the heights of all points on the surface. This altitude, sometimes referred to as a "sea level" or zero-level elevation , serves equivalently as a reference for
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